Pointless Parts of Completely Regular Locales

arXiv:2305.00096v1 [math.GN] 28 Apr 2023 POINTLESS PARTS OF COMPLETELY REGULAR FRAMES RICHARD N. BALL Dedicated to the memory of Bernhard Banaschewski, inspiration and friend. A BSTRACT. (Completely regular) locales generalize (Tychonoff) spaces; indeed, the passage from a locale to its spatial sublocale is a well understood coreflection. But a locale also possesses an equally important pointless sublocale, and with morphisms suitably restricted, the passage from a locale to its pointless sublocale is also a coreflection. Our main theorem is that every locale can be uniquely represented as a subdirect product of its pointless and spatial parts, again with suitably restricted projections. We then exploit this representation by showing that any locale is determined by (what may be described as) the placement of its points in its pointless part. 1. I NTRODUCTION The primary motivation for point free topology comes from classical point set topology. When expressed in terms of the underlying frames, the connection between these two worlds is the well known functor σ : F → sF, which assigns to each (completely regular) frame L its spatial part σL. (Here F is the category of completely regular frames with frame homomorphisms, and sF is its full subcategory of spatial frames, i.e., frames in which every element is the meet of the maximal elements above it.) The functor σ is an epireflection, and the reflector for a given frame L is the surjection   ^ ^ ↑b↑max L , ↑b↑max L = b 7→ b ∈ L. σL : L → σL ≡ b ∈ L : b = One of the principal advantages of the point free approach to general topology is its increase in extent beyond the spatial situation, and this article provides evidence of the benefits of that generality. We call a frame Date: April 28, 2023. 2020 Mathematics Subject Classification. Primary 06D22; Secondary 54C45, 54G12, 54G10. Key words and phrases. completely regular frame, compact coreflection, round filter. 1 2 R. N. BALL pointless if it has no maximal elements, and we show that every frame L has a pointless part πL which plays a role roughly complementary to its spatial part. We need to restrict the homomorphisms slightly to those we term skinny, and work with the restricted cateogory Fs of frames with skinny morphisms and its full subcategory of pointless frames plFs. (When the domain and codomain are spatial, the skinny frame homomorphisms are those whose pointed continuous functions have scattered fibers.) In that context the functor π : Fs → plFs is an epireflection, and a reflector for the frame L is the surjection   ^ { } ↑b↑ πL : L → πL ≡ a ∈ L : ∀b > a ∃c (b > c > a) = b 7→ πL , b ∈ L. Though disjoint, the two sublocales σL and πL are not complementary. Nevertheless, their reflectors σL and πL are diagnostic when taken together, in the sense that every frame L has a unique representation as a subdirect product of its pointless and spatial parts, with suitably restricted projection maps. This is our main Theorem 5.2.2. The targets of this representation, here termed fat, form a monoreflective subcategory of Fs, and even though we would like to know more about these objects (cf. Question 5.4.1), Theorem 6.3.3 provides a fairly concrete description of the reflector arrow. Thus a frame L is determined by the interaction of its spatial and pointless parts, and this interaction is governed by two principal connections. The first of these is the arrow λL : πL → πσL induced by applying the π functor to the σL arrow. σL L → σL ← ← ← ← πL → πσL → πL → πσL λL We refer to λL as the ligature of L (Subsection 5.1). The second principal connection between the spatial and pointless parts of a frame is motivated by the crucial observation that  each maximal element a of a frame L is associated with the filter ya ≡ b ∈ πL : b  a on πL. Such filters have two key features: they are round, i.e., for each b ∈ ya W there exists an element c ∈ ya such that c ≺≺ b, and ya b∗ = ⊤. We call such filters regular, and we show that each regular filter on a pointless frame E arises as ya for some frame L having pointless part E and maximal element a. Furthermore, the filters produced by distinct maximal elements are independent, i.e., contain disjoint elements of πL. We refer to the family POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 3 W ≡ { ya : a ∈ max L } of such filters as the spatial support of L. Together with the pointless part of L, W determines the fat reflection of an atomless frame (Theorem 6.3.3). W also determines whether or not L is spatial, i.e., whether σL is an isomorphism (Proposition 7.1.2), and whether or not L is compact (Proposition 7.1.4). In Section 8 we take up the situation that arises when all of the filters of the spatial support W are maximal proper round filters. This assumption has the great advantage that it guarantees the complete regularity of the synthetic construction of a frame from its pointless part E and its spatial support W. However, it has the disadvantage that the pointless part can grow bigger than E, at least if W is taken to be the entire family of maximal proper round filters on E. In fact, in the latter case we show in Proposition 8.1.5 that the synthetically constructed frame is βE, the compact coreflection of E. That fact motivates a final digression, in which we establish the existence of what in spatial terms is a particularly exotic compactification in compact Tychonoff spaces without isolated points (Corollary 8.3.6 and Proposition 8.3.1.) 2. P RELIMINARIES For purposes of handy reference and to fix notation, we record here the few background results which underlie what follows. The material is almost entirely folklore and should be skipped upon a first reading, and then consulted only as necessary. 2.1. Naked frames, completely regular frames, and witnessing families. The context for these remarks is the category F of completely regular frames with frame homomorphisms. Unless otherwise explicitly stipulated, all frames are assumed completely regular and all spaces are assumed Tychonoff. A couple of constructions require an excursion into the category nF of naked frames, i.e., frames without the hypothesis of complete regularity, and then a return to F by way of the completely regular coreflection. That is, we first construct a naked frame, labeled for instance L ′ , and then extract its largest completely regular subframe, labeled for instance L. The issue of whether a particular naked frame is completely regular thus plays an important role in what follows, and our working definition of this important notion is as follows. When speaking of two elements ai of a frame L, to say that a1 lies completely below a2 is to say that there is a family of witnesses {bp }Q ⊆ L such that a1 6 bp ≺ bq 6 a2 for all p < q in Q. We write a1 ≺≺ a2 , and we refer to {bp }Q as a witnessing family for a1 ≺≺ a2 . 4 R. N. BALL 2.2. Nuclei and prenuclei. Notation (nucleus notation). We make use of the following notational conventions for nuclei. • We use lower case Greek letters to denote nuclei on frames, often subscripted with the name of the frame, as in δL . • We use primed lower case Greek letters to denote prenuclei on frames, often subscripted with the name of the frame, as in δL′ . The corresponding nucleus is then designated by the same Greek letter without the prime. See Lemma 2.2.1. • We define and denote the kernel of a nucleus δ on a frame L to be ker δ ≡ { a ∈ L : δ(a) = ⊤ } . See Lemma 2.2.3. • For a nucleus δ on a frame L, we denote its fixed point set, aka its sublocale, by fix δ ≡ { a ∈ L : δ(a) = a } . Lemma 2.2.1. For a prenucleus δ ′ on a frame L, define for all a ∈ L and for all ordinals β: δβ (a) ≡ a, if β = 0, δβ (a) ≡ δ ′ ◦ δα (a) _ δβ (a) ≡ δα (a) if β = α + 1, if β is a limit ordinal, α<β δ(a) ≡ δβ (a) for some (any) β such that δβ (a) = δβ+1 (a). Then δ is the unique nucleus on L such that fix δ ′ = fix δ. Proof. See [16, III 11.5].  Lemma 2.2.2. For a filter F on a frame L, the map ! _ δF′ : L → L ≡ a 7→ (b → a) , F functions as a prenucleus on L. a ∈ L, POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 5 Proof. It is clear that a 6 δF′ (a) 6 δF′ (b) for a 6 b in L. And for elements ai ∈ L we have _ _ δF′ (a1 ∧ a2 ) = (b → (a1 ∧ a2 )) = ((b → a1 ) ∧ (b → a2 )) > _ F F F _ (a1 ∧ (b → a2 )) = a1 ∧ (b → a2 ) = a1 ∧ δF′ (a2 ).  F Definition (normal filter). A filter F on a frame L is said to be normal if δF (a) ∈ F implies a ∈ F for all a ∈ L. Here δF is the nucleus associated per Lemma 2.2.1 with the prenucleus δF′ of Lemma 2.2.2. Lemma 2.2.3. A filter on a frame is the kernel of a nucleus if and only if it is normal. In detail, if δ is a nucleus on a frame L then ker δ is a normal filter, and if F is a normal filter on L then δF is the unique nucleus on L for which F = ker δF . Proof. If F = fix δ for a nucleus δ and if δ(a) ∈ F then δ ◦ δ(a) = ⊤, and since δ is idempotent we get δ(a) = ⊤ and a ∈ F. On the other hand, suppose that F is a normal filter and δF′ is the prenucleus defined from it as in Lemma 2.2.2 and δα is defined from δF′ as in Lemma 2.2.1. Then since for any a ∈ F we have _ δF (a) > δ1F (a) = δF′ (a) = (b → a) = ⊤, F we see that F ⊆ ker δF . And for any a ∈ ker δF we have that a ∈ F since δF (a) = ⊤ ∈ F and F is normal.  Lemma 2.2.4. Let m and n be frame homomorphisms with common domain L, and suppose that m is surjective. Then n factors through m if and only if m(a) = ⊤ implies n(a) = ⊤ for all a ∈ L. ← → → n m ← L → M ← N Proof. Consider elements ai ∈ L such that m(a1 ) = m(a2 ). Find a subset W B ⊆ L such that B = a1 , and such that for each b ∈ B we have b ≺ a1 as witnessed by another element cb , i.e., cb ∨ a1 = ⊤ and cb ∧ b = ⊥. Then the fact that m(cb ∨ a2 ) = m(cb ∨ a1 ) = ⊤ implies that n(cb ) ∨ n(a2 ) = n(cb ∨ a2 ) = ⊤, combined with the fact that n(cb ) ∧ n(b) = n(cb ∧ b) = ⊥, W W yields n(b) ≺ n(a2 ). In sum we have n(a1 ) = n ( B) = B n(b) 6 n(a2 ), and a symmetrical argument gives n(a2 ) 6 n(a1 ).  6 R. N. BALL Lemma 2.2.5. Let m : L → M be a frame homomorphism, and let δL and δM be nuclei on L and M, respectively. Then m drops through δL and δM , i.e., there exists a unique map m̄ such that δM ◦ m = m̄ ◦ δL , if and only if m(ker L) ⊆ ker M. m L → M ← ← ← ← fix δL → δM → δL → fix δM m̄ Proof. This follows immediately from Lemma 2.2.4  2.3. Congruences. Notation (congruence notation). We make use of the following notational conventions for frame congruences. • We use capital Greek letters to denote congruences on a frame L, and we denote the congruence frame itself, aka the assembly of L, by con L. • (Φa , oa , Oa ) We denote the congruence of the open quotient associated with an element a of a frame L by Φa ≡ { (a1 , a2 ) : a ∧ a1 = a ∧ a2 } , and the quotient map by oa : L → L/Φa . We denote the quotient frame L/Φa by Oa , and often identify it with its sublocale {a → b : b ∈ L}, in which case we also identify the map oa with (the range restriction of) its nucleus (b 7→ a → b), b ∈ L. • We denote the congruence of the closed quotient associated with an element a of a frame L by Ψa ≡ { (a1 , a2 ) : a ∨ a1 = a ∨ a2 } , and the quotient map by ca : L → L/Ψa . We denote the quotient frame L/Ψa by Ca , and often identify it with its sublocale ↑a↑ ≡ { b : b > a }, in which case we also identify the map ca with (the range restriction of) its nucleus (b 7→ a ∨ b), b ∈ L. • We denote the coarsest dense congruence on a frame L by ∆L ≡ { (a1 , a2 ) : a∗1 = a∗2 } , and the quotient map by δL : L → L/∆L ≡ ∆L. We often identify the quotient with its sublocale L∗∗ ≡ {a∗∗ }L , in which case we also POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 7 identify the map δL with (the range restriction of) its nucleus (a 7→ a∗∗ ), a ∈ L. We refer to the map or it codomain as the skeleton of L. • We denote the congruence of a frame surjection m : L → M by Θm ≡ { (a1 , a2 ) : m(a1 ) = m(a2 ) } . Lemma 2.3.1. For elements a 6 b and congruence Ξ on a frame L, (a, b) ∈ Ξ ⇐⇒ Φa ∧ Ψb 6 Ξ. In particular, (a, ⊤) ∈ Ξ ⇐⇒ Φa 6 Ξ and (⊥, b) ∈ Ξ ⇐⇒ Ψa 6 Ξ. Lemma 2.3.2. For any frame surjection m : L → M, _ _ Θm > Φa ∨ Ψa . m−1 (⊤) m−1 (⊥) Proof. If a ∈ m−1 (⊤) and (a1 , a2 ) ∈ Φa then m(a1 ) = m(a1 ) ∧ ⊤ = m(a1 ) ∧ m(a) = m(a1 ∧ a) = m(a2 ∧ a) = m(a2 ) ∧ m(a) = m(a2 ) ∧ ⊤ = m(a2 ).  Lemma 2.3.3. Let m : L → M be a frame surjection. (1) The map m−1 : con M → ↑Θm ↑con L = ∆ 7→ m−1 (∆) ≡ { (a1 , a2 ) : (m(a1 ), m(a2 )) ∈ ∆ } is a frame isomorphism. (2) For any element b ∈ M, m−1 (Φb ) = Θm ∨ Φm∗ (b) .
Proof. (2) To show that Θm ∨Φm∗ (b) 6 m−1 (Φb ), first note that for elements ai ∈ L such that (a1 , a2 ) ∈ Θm we have m(a1 ) = m(a2 ) =⇒ m(a1 ) ∧ b = m(a2 ) ∧ b ⇐⇒ (m(a1 ), m(a2)) ∈ Φb ⇐⇒ (a1 , a2 ) ∈ m−1 (Φb ). Then observe that for any elements ai ∈ L such that (a1 , a2 ) ∈ Φm∗ (b) we also have a1 ∧ m∗ (b) = a2 ∧ m∗ (b) =⇒ m(a1 ) ∧ b = m(a1 ) ∧ m ◦ m∗ (b) = m(a2 ) ∧ m ◦ m∗ (b) = m(a2 ) ∧ b ⇐⇒ (m(a1 ), m(a2)) ∈ Φb ⇐⇒ (a1 , a2 ) ∈ m−1 (Φb ). 8 R. N. BALL To show that Θm ∨ Φm∗ (b) > m−1 (Φb ), observe that for any pair of elements ai ∈ L such that (a1 , a2 ) ∈ m−1 (Φb ) we have (m(a1 ), m(a2 )) ∈ Φb ⇐⇒ m(a1 ) ∧ b = m(a2 ) ∧ b =⇒ m∗ ◦ m(a1 ) ∧ m∗ (b) = m∗ ◦ m(a2 ) ∧ m∗ (b) ⇐⇒ (m∗ ◦ m(a1 ), m∗ ◦ m(a2 )) ∈ Φm∗ (b) . Since (ai , m∗ ◦m(ai )) ∈ Θm , we can conclude from transitivity that (a1 , a2 ) ∈ Θm ∨ Φm∗ (b) .  2.4. Successors and predecessors, atoms and maximal elements. Definition (successor, predecessor, a+ , atom, maximal element, max L, pointless frame, interpolative lattice). When speaking of two elements a and c of a distributive lattice L, we say that c is a successor of a, or that a is a predecessor of c, if c > a and for any element b such that a 6 b 6 c, either b = a or b = c. We denote the set comprised of a together with its successors by a+ ≡ {a} ∪ { c : c is a successor of a } . A successor of ⊥ is called an atom of L, and an element having ⊤ as a successor is called maximal. We denote the set of maximal elements of a frame L by max L ≡ { a : a is maximal in L } . A frame L is called pointless if max L = ∅. Thus the “empty frame” {⊥ = ⊤}, i.e., the topology of the empty space, is pointless, whereas the two element frame 2 = {⊥ = 6 ⊤}, i.e., the topology of the singleton space, is not. A lattice L is called interpolative if it has no successors or predecessors, i.e., if for all elements a < c there exists an element b such that a < b < c. Lemma 2.4.1. The following hold for elements a and b in a distributive lattice L. (1) The maps [a ∧ b, a] ∋ c −→ c ∨ b and c ∧ a ←− c ∈ [b, a ∨ b] are inverse lattice isomorphisms. (2) a ∧ b+ ⊆ (a ∧ b)+ . Proof. (2) Consider a successor c of b. Since b 6 (a ∧ c) ∨ b = (a ∨ b) ∧ c 6 c, POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 9 either (a ∧ c) ∨ b = b or (a ∨ b) ∧ c = c. The first possibility implies that a ∧ c 6 b and hence a ∧ c = a ∧ b, while the second implies that a ∨ b > c and hence that a ∧ c is a successor of a ∧ b by part (1). In either case we get that a ∧ c ∈ (a ∧ b)+ .  Lemma 2.4.2. Let m : L → M be a frame homomorphism. (1) If b is prime (aka meet irreducible) in M then m∗ (b) is prime in L. (2) If m is surjective then m∗ (max M) ⊆ max L. (3) If m is surjective and a is prime in L then m(a) is prime in M. Proof. (3) Let a be a prime element of L such that m(a) < ⊤, and consider elements ci ∈ M such that c1 ∧ c2 = m(a). Let ai ≡ m∗ (ci ), i = 1, 2, and note that a1 ∧ a2 = m∗ (c1 ) ∧ m∗ (c2 ) = m∗ (c1 ∧ c2 ) = m∗ ◦ m(a) > a Now a is maximal in L because L is regular, so m∗ ◦ m(a) is either a or ⊤. The latter possibility is ruled out by the fact that m ◦ m∗ ◦ m(a) = m(a), so we have a1 ∧ a2 = a. By the primeness of a, then, either a1 6 a or a2 6 a, hence m(a1 ) 6 m(a) or m(a2 ) 6 m(a). But since m is surjective, m(ai ) = m ◦ m∗ (ci ) = ci , so either c1 6 m(a) or c2 6 m(a), i.e., m(a) is prime in M.  Lemma 2.4.3 explains, among other things, how complemented successor elements arise from maximal elements in frame factors. Lemma 2.4.3. The following hold in a frame L. (1) An element is a predecessor of ⊤ if and only if it is maximal if and only if it is prime and unequal to ⊤. (2) An element of a frame is an atom if and only if it is the complement of a maximal element. (3) An element a is a successor of an element b if and only if a > b and a → b is a maximal element of L. (4) An element b is a predecessor of an element a if and only if b = a ∧ c for a maximal element c  a. (5) If L = M × N and a is a maximal element of M then (⊤, ⊥) is a successor of (a, ⊥) and (⊤, ⊥) has complement (⊥, ⊤). In this case (⊤, b) is a successor of (a, b) for all b ∈ N. (6) Every complemented successor in L arises as in (3). That is, if c is a successor of a in L and c is complemented then the map m : L → ↓c↓ × ↓c∗ ↓ = (b 7→ (b ∧ c, b ∧ c∗ )) 10 R. N. BALL is an isomorphism, and m(c) = (c, ⊥) is a successor of m(a) = (a, ⊥). Proof. The assumption that L is regular implies the equivalence of maximality and primeness in (1). Likewise in (2), since an atom a ∈ L is the join of a set A of elements rather below it, there must be at least one element b ∈ A such that b > ⊥. This implies that b = a, but more to the point, that a is rather below itself, hence a is complemented. (3) If a is a successor of b then b is maximal in ↓a↓, and since the map ↓a↓ → Oa = (c 7→ a → c) is a frame isomorphism, a → b is maximal in Oa . Thus a → b is prime in Oa by (1), and oa∗ (a → b) = a → b is prime in L by Lemma 2.4.2(1). On the other hand, if a > b and a → b ∈ max L then a∨(a → b) = ⊤ since a  a → b. Therefore Lemma 2.4.1(1) provides an isomorphism between the intervals [b, a] and [a → b, ⊤], from which we see that a is a successor of b.  Lemma 2.4.4. For any frame L, max(con L) = { Ψb : b ∈ max L }. Consequently the set of atoms of con L is {Φb }max L . Proof. For an element b ∈ max L, consider a pair (a1 , a2 ) ∈ / Ψb , so that ai  b > aj for i 6= j, say a1  b > a2 . Then any congruence which contains both Ψb and (a1 , a2 ) must contain every pair (a3 , a4 ). For if (a3 , a4 ) ∈ / Ψb , say a3  b > a4 , then Ψb contains both (a1 , a3 ) and (a2 , a4 ), hence also (a3 , a1 ) by symmetry, and then finally (a3 , a4 ) by transitivity. We have proven that Ψb ∈ max(con L). On the other hand, suppose Ξ ∈ max(con L). Then L/Ξ is isomorphic to W 2 ≡ {⊥, ⊤}, and if q : L → 2 is the quotient map then b ≡ q−1 (⊥) ∈ max L is such that Ξ = Ψb . The second sentence of the lemma follows from the first via Lemma 2.4.3(2).  Lemma 2.4.5. A frame L is pointless if and only if it is interpolative. Proof. An interpolative frame is clearly pointless. And in any frame L, if elements a < c admit no element b such that a < b < c then a is maximal in ↓c↓, and since ↓c↓ ≈ Oc = {c → b}L , it follows that c → a is maximal in Oc . Consequently oc∗ (c → a) = c → a is maximal in L by Lemma 2.4.2(1).  2.5. Round and regular filters. Definition (filter, independent family of filters, round filter, regular filter, rflt L, round ideal). A filter on a frame L is a nonempty upset F ⊆ L which POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 11 is closed under binary meets. The filter is said to be proper if it does not contain ⊥. A family of filters is said to be independent if any two distinct filters of the family contain disjoint elements. A filter F is said to be round if for every element a ∈ F there exists an element b ∈ F such that b ≺≺ a. Round ideals are defined dually. A filter F is said to be regular if it is round W and F b∗ = ⊤. We denote the family of proper regular filters on a frame L by rflt L. Notation (xa ). We use lowercase letters near the end of the Latin alphabet, e.g., x, y, z, to designate filters on a frame. For a maximal element a, we  denote the filter b : b  a by subscript, e.g., xa , ya , za . We use uppercase letters near the end of the alphabet, e.g., W, X, Y, Z, to designate families of filters on a frame. Lemma 2.5.1 records the basic information about round and regular filters. Lemma 2.5.1. The following hold in any frame L. (1) Every proper round filter is contained in a maximal proper round filter. (2) For every proper filter x ⊆ L, x̊ ≡ { a ∈ x : ∃b ∈ x (b ≺≺ a) } (3) (4) (5) (6) (7) (8) (9) is the largest round filter contained in x. A proper round filter x is maximal if and only if for every element a∈ / x and every element b ≺≺ a there exists an element c ∈ x such that b ∧ c = ⊥. Distinct maximal proper round filters contain disjoint elements. Therefore any family of maximal proper round filters is independent. W If x is a maximal proper round filter then x b∗ is a prime element of W ∗ W L. That is, either x is regular, i.e., x b = ⊤, or x b∗ = a ∈ max L. In the latter case x = xa . If a ∈ max L then xa is a maximal proper round filter such that W ∗ xa b = a. W For any a ∈ max L, xa is completely prime, i.e., A ∈ xa implies A ∩ xa 6= ∅ for any subset A ⊆ L. A proper round filter is maximal if and only if it is of the form ẙ for some ultrafilter y on L. A maximal proper round filter x on L has the following sort of primeness: (ai ≺≺ bi and a1 ∨ a2 ∈ x) =⇒ (b1 ∈ x or b2 ∈ x). 12 R. N. BALL Conversely, any proper round filter having this sort of primeness is maximal (among proper round filters). (10) The maps x −→ ha∗ : a ∈ xiidl and hb∗ : b ∈ Iifltr ←− I constitute inverse order preserving bijections between the sets of round filters and round ideals on L. Proof. (3) Any proper round filter x satisfying this condition is clearly maximal, and if a proper round filter x and element a ∈ / x violate this condition, i.e., if there exists an element b ≺≺ a such that b ∧ c > ⊥ for all c ∈ x, then the filter y generated by x ∪ {b} has the feature that ẙ is a proper round filter properly containing x. (4) If maximal round filters xi are distinct then there exist elements ai ∈ xi r xj and elements bi ∈ xi such that bi ≺≺ ai . By (3) there exist elements ci ∈ xi such that ci ∧bj = ⊥. But then ci ∧bi ∈ xi and (c1 ∧b1 ) ∧(c2 ∧b2 ) = ⊥. (5) Suppose that x is a maximal proper round filter, and for the sake of W argument suppose that x b∗ = a < c < ⊤. We claim that c ∈ / x, for otherwise there exists an element b ∈ x such that b ≺≺ c. But if d witnesses b ≺ c, i.e., if b ∧ d = ⊥ and c ∨ d = ⊤, then we get d 6 b∗ 6 a 6 c, resulting in the contradiction c = c ∨ d = ⊤. Since a < c there exists an element d ≺≺ c such that d  a, whereupon part (3) above produces an element b ∈ x such that b ∧ d = ⊥. We are led W to the contradiction d 6 b∗ 6 a. We conclude that x b∗ is prime. If b ∈ x then since x is round there exists an element c ∈ x such that c ≺ b. Let d witness c ≺ b, i.e., c ∧ d = ⊥ and d ∨ b = ⊤, so that d 6 c∗ 6 a. Then b  a, for otherwise ⊤ = d ∨ b 6 a, contrary to assumption. On the other hand, if b ∈ / x then every element c ≺≺ b is disjoint from some element W d ∈ x, hence c 6 d∗ 6 a. It follows that b = { c : c ≺≺ b } 6 a. W (6) If a ∈ max L and b ∈ xa then, since b = C for C ≡ { c : c ≺≺ b }, it must be the case that C ∩ xa 6= ∅, for otherwise the fact that C ⊆ ↓a↓ would W lead to the contradiction b = C ∈ ↓a↓. Thus xa is round. To see that xa is maximal among proper round filters, consider elements d ≺≺ c ∈ / xa . Then c 6 a, so there exists an element b witnessing d ≺ a, i.e., b ∧ d = ⊥ and b ∨ a = ⊤. Then b  a, for otherwise we would have a = b ∨ a = ⊤, POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 13 W contrary to hypothesis. That is, b ∈ xa . The fact that xa b∗ = a now follows from (5). W (7) If for some subset A ⊆ L and element a ∈ max L we have A = b ∈ xa W then A * ↓a↓, for otherwise b = A 6 a, contrary to assumption. (8) If a proper round filter x is maximal then it is the largest round filter contained in any filter containing it. And if x = ẙ for some ultrafilter y on L and if b ≺≺ a ∈ / x then there must exist an element b1 such that b ≺≺ b1 ≺≺ a, for which we know that b, b1 ∈ / y by (3). By virtue of the ∗ ∗ maximality of y, therefore, b , b1 ∈ y, and since b∗1 ≺≺ b∗ we get b∗ ∈ x by (2). We can conclude that x is maximal among proper round filters by (3). (9) Let x be a maximal proper round filter and let y be an ultrafilter for which x = ẙ, and suppose that ai ≺≺ bi . If a1 ∨ a2 ∈ x then a1 ∨ a2 ∈ y, hence either a1 ∈ y or a2 ∈ y, with the result that either b1 ∈ ẙ = x or b2 ∈ ẙ = x. Conversely, suppose a proper round filter x enjoys this sort of primeness, and consider elements b ≺≺ a ∈ / x with witnessing family {cp }Q for b ≺≺ a. Fix p < q < r in Q. Then c∗q ∨ cr = ⊤ ∈ x, and since c∗q ≺≺ c∗p with witnessing family {c∗s }p<s<q and cr ≺≺ a with witnessing family {cs }r<s , the primeness of x yields c∗p ∈ x or a ∈ x, and the latter condition is ruled out by assumption. We conclude that x is maximal among proper round filters by (3). (10) This follows from the fact that for elements a and b in any frame, a ≺≺ b if and only if b∗ ≺≺ a∗ .  An immediate consequence of Lemma 2.5.1(5) will be important in what follows. Corollary 2.5.2. A maximal proper round filter on a pointless frame is regular. Lemma 2.5.3. Let m : L → M be a frame surjection. (1) If x is a round filter on L then m(x) generates a round filter on M. (2) The filter generated by m(x) is proper if and only if m∗ (⊥) ∈ / x. (3) If x is a maximal proper round filter on L and y is a maximal proper round filter on M containing m(x) then m−1 (y)˚= x. (4) If x is a regular filter on L and m is dense then m(x) is a regular filter on M. (5) If x = xa for some a ∈ max L such that m(a) = ⊤, and if m is dense, then m(x) is a regular filter on M. 14 R. N. BALL (6) (Converse of (5)) If y is a regular filter on a frame M then there is a frame L admitting a dense surjection m : L → M and having a maximal element a ∈ max L such that m(xa ) = y and m(a) = ⊤. Proof. (4) If ⊤ > a ∈ M then ⊤ > m∗ (a) ∈ L, and since x is regular there exists an element b ∈ x such that b∗  m∗ (a). It follows that m(b∗ )  a, and since m is dense, m(b∗ ) = m(b)∗ . (5) Consider an element ⊤ > c ∈ M. Then ⊤ > m∗ (c) ∈ L, and we claim W that m∗ (c)  b∗ for some b ∈ xa . For otherwise m∗ (c) > xa b∗ = a by parts (5) and (6) of Lemma 2.5.1, in which case we would have c = m ◦ m∗ (c) > m(a) = ⊤, contrary to assumption. As before, it follows from the claim that m(b∗ )  c, and also as before m(b∗ ) = m(b)∗ because m is dense. (6) This is essentially the content of [2, 4.2.1]. Unfortunately, the result is incorrect as its stands, inasmuch as the crucial hypothesis of roundness is omitted from the definition of regular filter.1 Fortunately, the error does not invalidate the subsequent results in that article, and with the aid of the missing hypothesis of roundness, the reader will have no difficulty supplying a correct proof.  Corollary 2.5.4. For any frame surjection m : L → M, the set {m(xa )}max L is an independent family of regular filters on M. Proof. The family {xa }max L is independent by Lemma 2.5.1(4). Since m is  dense, {m(xa )}max L is independent as well. 3. T WO NUCLEI In the section we introduce and investigate the nuclei of interest. 3.1. The nucleus σ and the spatial part of a frame. Lemma 3.1.1. On a frame L, the map   ^ ↑a↑max L σL : L → L ≡ a 7→ is a nucleus with fixed point set fix σL = and kernel a:a= ^ ↑a↑max L ≡ σL ker σL = { a : ↑a↑max L = ∅ } . 1 The error in the proof occurs in the second paragraph, starting with the sentence which begins “On the other hand it is obvious that . . . ” POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 15 Proof. The map σL is clearly increasing and order preserving; to verify that it preserves binary meets, simply compute ^ ^ ↑a1 ↑max L ∧ ↑a2 ↑max L = σL (a1 ) ∧ σL (a2 ) = ^
^ ↑a1 ↑max L ∧ ↑a2 ↑max L = ↑a1 ∧ a2 ↑max L = σL (a1 ∧ a2 ). The penultimate equality is attributable to the primeness of maximal elements.  Recall that a frame is called spatial if every element is the meet of the maximal elements above it. See [16, II 5.3]. Proposition 3.1.2. The following hold for any frame L. (1) σL is spatial. (2) L is spatial if and only if σL is an isomorphism if and only if ker σL = {⊤} if and only if ∀a < ⊤ ∃b ∈ max L (a 6 b). (3) L is pointless if and only if σL is empty, i.e., σL = {⊤}. Proof. The nucleus σL fixes each maximal element, and each maximal element of L remains maximal in σL.  Definition (spatial part of a frame, sF). For a frame L, we refer to the map σL : L → σL or to its codomain σL as the spatial part of L. We designate the full subcategory of F comprised of the spatial frames by sF. Lemma 3.1.3. Any frame homomorphism m : L → M drops through σL and σM , i.e., there is a unique frame homomorphism m̄ such that σm ◦ m = m̄ ◦ σL . → σM → σL ← σL → M ← m ← ← L . → σM m̄ Proof. According to Lemma 2.2.5, it is sufficient to show that m(a) ∈ ker σM for any a ∈ ker σL . But this is clearly so, for if m(a) ∈ / ker σM = { b ∈ M : ↑b↑max M = ∅ } it is only because m(a) > b for some b ∈ max M, in which case m∗ (b) ∈ ↑a↑max L , contrary to assumption.  Proposition 3.1.4. sF is epireflective in F, and a reflector for the frame L is its spatial part σL : L → σL. 16 R. N. BALL Proof. This follows directly from Lemma 3.1.3.  3.2. The nucleus π and the pointless part of a frame. Proposition 3.2.1. In a frame L, the map _ πL′ (a) ≡ a+ is a prenucleus, with fixed point set πL ≡ { a : a has no successor } = { a : ∀c > a ∃b (a < b < c) } , and with kernel being the normal filter generated by max L, namely ker πL = { a : ∀b > a (b < ⊤ =⇒ b has a successor) } . Proof. The map πL′ is increasing by construction. To show that πL′ is orderpreserving, consider a 6 b in L. Then for each c ∈ a+ , either c 6 b or b ∨ c ∈ b+ by Lemma 2.4.1, with the result that _ _ πL′ (a) = a+ 6 b+ = πL′ (b). To check that πL′ (a ∧ b) > a ∧ πL′ (b), simply observe that by Lemma 2.4.1(2) we have _ _ _ πL′ (a ∧ b) = (a ∧ b)+ > (a ∧ b+ ) = a ∧ b+ = a ∧ πL′ (b). Certainly max L ⊆ ker πL , i.e., πL (a) = ⊤ for each a ∈ max L; after all, ⊤ ∈ a . This establishes that δmax L 6 πL , where δmax L represents the nucleus corresponding as in Lemmas 2.2.1 and 2.2.2 to the filter generated by max L. We claim that δmax L (a) > c whenever c is a successor of a. For in that case b ≡ c → a ∈ max L by Lemma 2.4.3(3), hence + ⊤ = δmax L (c → a) 6 δmax L (c) → δmax L (a) =⇒ c 6 δmax L (c) 6 δmax L (a). The claim implies that πL 6 δmax L .  The fact that ker πL is generated by the maximal elements of L (Proposition 3.2.1) implies that the congruence of πL is the join of the open congruences of the elements of max L. Corollary 3.2.2. For any frame L, ΘπL = _ Φa . max L POINTLESS PARTS OF COMPLETELY REGULAR FRAMES In terms of sublocales, πL = fix πL = \ 17 Oa . max L Definition (pointless part of a frame). For a frame L, we refer to the map πL : L → πL or its codomain as the pointless part of L. The pointless part of a frame can be characterized in terms of its maximal elements. Proposition 3.2.3. For any frame L, πL = { a : ∀c (a 6 c ∈ max L =⇒ c → a = a) } . Proof. Suppose that a does not lie in the set displayed on the right, say a 6 c for some c ∈ max L such that c → a > a. Then c → a is a successor of a by Lemma 2.4.3(3) because (c → a) → a = c by virtue of the maximality of c. Therefore πL (a) > c → a > a so a ∈ / πL. On the other hand, if a∈ / πL then a has a successor c, c → a ∈ max L by Lemma 2.4.3(3), and (c → a) → a > c > a.  Proposition 3.2.3 can be reformulated to provide a first order condition for membership in πL. Proposition 3.2.4. An element a in a frame L lies in πL if and only b → a is not prime for any b > a. That is, a ∈ πL if and only if
∀b > a ∃ci b ∧ c1 ∧ c2 6 a but (b ∧ c1  a and b ∧ c2  a) . Notation (punctured element, unpunctured element). Let us agree to call an element a of a frame L punctured if it has a successor in L, and unpunctured if not.2 Thus an element a is punctured if and only if it is of the form c ∧ b for some b > a and c ∈ max L. Otherwise put, a is punctured if and only if there exists some b > a for which b → a ∈ max L. 3.3. An example: the pointless real numbers πOR. Consider the topology OR of the real numbers, whose skeleton δOR is the complete and atomless boolean algebra of regular open subsets of R. An open subset of the reals is uniquely expressible as a disjoint union of open intervals, and such a set is unpunctured if and only if no pair of its intervals are abutting, i.e., share an endpoint. A familiar example of an unpunctured open set is the complement of the Cantor set. The sublocale of unpuntured subsets of the reals forms a sublocale which evidently strictly contains the skeleton. 2 This evocative terminology was suggested to the author by Fred Dashiell ([11]). 18 R. N. BALL p, ↼ q}Q and the following Proposition 3.3.1. πOR is the frame with generators {⇀ relations indexed by the rational numbers p, q ∈ Q. (1) ⇀ p∨↼ q = ⊤ if p 6 q. ⇀ q = ⊥ if p > q. (2) p ∧ ↼ W W ⇀ (3) p = p<q ⇀ q and ↼ q = p<q ↼ p for all p and q. W ↼ V ⇀ (4) Q q = ⊥ and Q p = ⊤. The classical presentation of the locale of real numbers ([16]) differs from the presentation in Proposition 3.3.1 in only one small detail: the 6 operation the first relation is replaced by the < operation in the presentation of the reals. However, any speculation that the (pointfree) Yosida adjunction ([15], [3]) connecting F with W has a pointless counterpart founders on the observation that the family of pointfree real valued localic functions on a frame lacks (the frame counterparts of the) constant functions, and in particular lacks a 0 function. (Here W is the category of archimedean lattice-ordered groups with designated weak order unit.) Curiously, the family does have a negation operation. 3.4. Decomposing a frame into its scattered and atomless parts. In classical topology a space is called scattered if every nonempty closed subset contains an isolated point. We take the frame counterpart of this notion as our definition, while acknowledging that another definition is the starting point of the extensive and scholarly treatment of the topic in Chapter IX of the excellent reference [17]. See also [7]. Definition (scattered frame, scattered element, coscattered element). A frame L is called scattered if every element a < ⊤ has a successor. An element a of a frame L is called scattered if ↓a↓L (or Oa ) is a scattered frame, and it is called coscattered if ↑a↑L (or Ca ) is a scattered frame. Lemma 3.4.1. The following hold in an arbitrary frame L with e ≡ πL (⊥). (1) e is the largest scattered element of L, and oe : L → Oe is the largest scattered open quotient of L. (2) L is scattered if and only if e = ⊤ if and only if ⊥ is coscattered if and only if the pointless part of L is empty. (3) e = ⊥ if and only if the pointless part πL is a dense quotient of L if and only if L is atomless. Proof. The condition that πL = {⊤} is equivalent to the condition that π∗L (⊥) = ⊤ by Proposition 3.2.1, which is to say that πL′ (a) > a for all a < ⊤.  POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 19 Definition (scattered and atomless parts of a frame). For a frame L, let e ≡ πL (⊥). We refer to the map oe : L → Oe or to its codomain as the scattered part of L. We refer to the map ce : L → Ce or to its codomain as the atomless part of L. Definition (subdirect product of frames). A subdirect product is a subobject of a product for which the projection morphisms are surjective. Proposition 3.4.2. Every frame is (isomorphic to) a subdirect product of its scattered and atomless parts. Its scattered part is an open quotient and its atomless part is the complementary closed quotient. The frame is scattered if and only if its atomless part is empty and atomless if and only if its scattered part is empty. Proof. We have the product map oe × ce : L → Oe × Ce , which is one-one because the corresponding congruences Φe and Ψe are complements in con L and therefore intersect to the identity congruence.  3.5. Decomposing a frame into its pointless and spatial parts. Lemma 3.5.1. For any frame L, σL ∧ πL = ⊥con L = 1L . Proof. We prove by induction that σL (a) ∧ πα L (a) = a for any a ∈ L and any ordinal α. For α = 0 it is clear that σL (a) ∧ π0L (a) = σL (a) ∧ a = a. Assume the assertion holds for all ordinals β < α. In case α = β + 1 we have _ _ β ′ σL (a) ∧ πα (a) = σ (a) ∧ π ◦ π (a) = σ (a) ∧ c = (σL (a) ∧ c). L L L L L + πβ (a)+ πβ L (a) L β + If c ∈ πβ then either c = πβ L (a) L (a) or c is a successor of πL (a). In the first case σL (a) ∧ c = a by the inductive hypothesis. In the second case b ≡ c → πβ max L by Lemma 2.4.3(3), and since b > πβ L (a) ∈ L (a) > a, it V follows that b > ↑a↑max L = σL (a). Therefore σL (a) ∧ c 6 b ∧ c 6 πβ L (a), β so that by the inductive hypothesis we get σL (a) ∧ c 6 σL (a) ∧ πL (a) = a. Since the persistence of the assertion through limit ordinals is evident, the induction is complete.  Whereas Lemma 3.5.1 establishes that the congruences associated with the pointless and spatial parts of a frame are disjoint, Lemma 3.5.2 points out that they are not complementary. Lemma 3.5.2. In any frame L, σπL is empty, whereas πσL need not be empty. Otherwise put, σL ◦ πL (a) = ⊤ for all a ∈ L, whereas πL ◦ σL (a) need not be ⊤ for all a ∈ L. 20 R. N. BALL Proposition 3.5.3. Every frame is (isomorphic to) a subdirect product of its pointless and spatial parts. The frame is pointless if and only if its spatial part is empty, and spatial if and only if its pointless part is empty. Proof. The subdirect representation is the product morphism of σL with πL . The product map is one-one by Lemma 3.5.1.  Proposition 3.5.4. For any frame L with e ≡ πL (⊥), we have the following commuting diagram in F. ← Ce πL ce ← ← → → ← ← σL oe → → → → Oe × Ce ← ← ← σL → ← L ← ← ← πL × σL → → → πL → → Oe Concerning the two squares which have L at one corner, the left square exhibits the decomposition of L into its pointless and spatial parts (Proposition 3.4.2), while the right square exhibits the decomposition of L into its scattered and atomless parts (Proposition 3.5.3). The top arrow exists because Ce is the closure of πL, and the bottom arrow exists because σL is the spatial reflector of L and Oe is spatial. Corollary 3.5.5. The pointless part of any frame L is isomorphic to the pointless part of its atomless part. In symbols, πL ≈ πCe . Proof. The elements of πL are the elements of L lacking successors in L, while the elements of πCe all the elements of Ce lacking successors in Ce . Since x x   πL ⊆ πL (⊥) = ↑e↑ = Ce , these two sets coincide.  We return to the topic of subdirect product decompositions of a frame in Subsection 5.2. 4. T HE CATEGORY Fs In Section 3 we took pains to emphasize the parallels between the spatial and pointless parts of a frame. But we must now acknowledge an important difference between the two: whereas the spatial part of a frame is reflective, POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 21 the pointless part is not. However, by restricting the frame homomorphisms we can make the pointless part reflective as well. 4.1. Skinny frame homomorphisms. Lemma 4.1.1. The following are equivalent for a frame homomorphism m : L → M. (1) m takes coscattered elements of L to coscattered elements of M. (2) m(ker πL ) ⊆ ker πM . (3) m(max L) ⊆ ker πM , i.e., πM ◦ m(a) = ⊤ for all a ∈ max L. (4) m drops through πL and πM , i.e., there exists a unique map m̄ such that πM ◦ m = m̄ ◦ πL . (5) m takes unpunctured elements of L to unpunctured elements of M. Proof. (2) certainly implies (3) because max L ⊆ ker πL , and (3) implies (2) because m−1 (ker πM ) is a normal filter containing max L and ker πL is the normal filter generated by max L. The equivalence of (2) and (4) is an application of Lemma 2.2.5.  Definition (skinny frame homomorphism, skinny contiinuous function). We refer to a frame homomorphism m : L → M which satisfies the conditions of Lemma 4.1.1 as being skinny. The spatial counterpart of the notion of a skinny frame homomorphism is a continuous function which inversely preserves closed scattered subsets, or equivalently, a continuous function with scattered fibers. We shall refer to such functions as skinny functions. Definition (Fs, plFs). The category Fs has objects which are frames and morphisms which are skinny frame homomorphisms. The full subcategory plFs is comprised of the pointless frames. Corollary 4.1.2 establishes that Fs is a legitimate category with plFs as an epireflective subcategory. Corollary 4.1.2. (1) A frame isomorphism is skinny. (2) A frame surjection is skinny. (3) The composition of skinny frame homomorphisms is skinny. (4) If e is a frame surjection and m is a frame homomorphism such that m ◦ e is skinny then m is skinny. (5) plFs is epireflective in Fs. Proof. (2) holds because Lemma 2.4.2(3) tells us that a frame surjection satisfies Lemma 4.1.1(3). (3) holds because the third property of Lemma 4.1.1 22 R. N. BALL is clearly preserved by composition. To check (4), consider frame homomorphisms e : L → K and m : K → M such that m ◦ e is skinny and e is surjective. If a is a maximal element of K then e∗ (a) ≡ b is a maximal element of L by Lemma 2.2.5(2), and m(b) = a. Since m ◦ e is skinny we get πM ◦ m(a) = πM ◦ e(b) = ⊤. (5) is a consequence of Lemma 4.1.1(4).  The diagram of Proposition 3.5.4 exists in Fs. Proposition 4.1.3. All the mappings in the diagram of Proposition 3.5.4 are skinny. Proof. The diagonal arrows are skinny by virtue of being surjective. The top and bottom arrows are skinny by Corollary 4.1.2(4). The arrows L → πL×σL and L → Oe ×Ce can both be shown to induce bijections on maximal elements, and are therefore both skinny.  4.2. A factorization structure for Fs. Lemma 4.2.1. The following are equivalent for a frame surjection e : L → M. The joins are taken in con L. W (1) Θe 6  Φa : a ∈ e−1 (⊤) ∩ (max L) . W Φa : a ∈ e−1 (⊤) ∩ (max L) . (2) Θe = (3) Θe is a join of atoms. Proof. The equivalence of (1) and (2) is a consequence of 2.3.2, while the equivalence of (2) and (3) follows from Lemma 2.4.4.  Definition (E, M). Let E be the class of Fs-surjections which satisfy the mi Mi )I conditions of Lemma 4.2.1. Let M be the class of all Fs-sources (L −→ such that for every a ∈ max L there exists an index i ∈ I such that mi (a) < ⊤. Proposition 4.2.2. Fs has (E, M)-factorization of sources. m i Mi )I in Fs. Let Proof. Consider a source S ≡ (L −→ 
P ≡ a ∈ max L : ∀i ∈ I mi (a) = ⊤ , W let Ξ ≡ P Φa , and denote the quotient map by e : L → L/Ξ ≡ b L. The map e is skinny by Corollary 4.1.2(2). Note that e(a) = ⊤ for each a ∈ P because (a, ⊤) ∈ Φa since a∧a = a∧⊤ and Φa ⊆ Ξ. It follows that e ∈ E. Note that Φa ⊆ Θmi for any a ∈ P and i ∈ I, for if (a1 , a2 ) ∈ Φa then a1 ∧ a = a2 ∧ a, and since mi (a) = ⊤ we get mi (a1 ) = mi (a1 ) ∧ mi (a) = mi (ai ∧ a) = mi (a2 ∧ a) = mi (a2 ). POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 23 It follows that Ξ ⊆ Θmi , so that mi factors through e, say mi = m̂i ◦ e. The maps m̂i are skinny by Corollary 4.1.2(4). m̂i Mi )I lies in M. Finally, we claim that the factored source b S ≡ (b L −→ b For if c is a maximal element of L then b ≡ e∗ (c) is a maximal element of W L such that (b, ⊤) ∈ / Ξ = P Φa . In particular, b ∈ / P because otherwise W (b, ⊤) ∈ Φb ⊆ P Φa = Ξ, contrary to assumption. It follows that there exists some index i ∈ I such that mi (b) < ⊤, which yields ⊤ > mi (b) = m̂i ◦ e(b) = m̂i (c). This proves the claim and the proposition.  Proposition 4.2.3. Fs has the (E, M)-diagonalization property for sources. m i Proof. Consider a commuting square in Fs with e ∈ E and (K −→ Mi )I ∈ M. K ← → → f → N ← ni mi→ ← d e → ← ← L Mi We claim that P ≡ e−1 (⊤) ∩ (max L) ⊆ f−1 (⊤). For if f(a) < ⊤ for some a ∈ e−1 (⊤) ∩ (max L) then since f(a) ∈ ker πK by Lemma 4.1.1(3), there mi Mi )I ∈ M, must be an element b ∈ max K such that b > f(a). Since (K −→ there must be an index i ∈ I for which mi (b) < ⊤ in Mi , which leads to the contradiction ⊤ > mi (b) > mi ◦ f(a) = ni ◦ e(a) = ni (⊤) = ⊤. The claim shows that Θe = ⊆ _ _ Φa : a ∈ e−1 (⊤) ∩ (max L) Φa : a ∈ f−1 (⊤) ∩ (max L) ⊆ Θf , with the result that the diagonal function d exists by Lemma 2.2.4. This map is skinny by Corollary 4.1.2(4).  Proposition 4.2.4. Fs is an (E, M)-category. Proof. Propositions 4.2.2 and 4.2.3 show that Fs has the features required by Definition 15.1 in [1].  4.3. A second look at the pointless reflection in Fs. The (E, M)-factorization structure allows us to refine our understanding of plFs as an epireflective subcategory of Fs (Corollary 4.1.2(5)). Theorem 4.3.3 shows that it is actually an E-reflective subcategory, and this distinction will become important in Subsection 5.2. 24 R. N. BALL Pointlessness has several pleasing characterizations in terms of E-morphisms and M-morphisms. Lemma 4.3.1. The following are equivalent for a frame M. (1) (2) (3) (4) Every frame homomorphism out of M has a pointless codomain. Every M-morphism into M has a pointless domain. Every E-morphism out of M is an isomorphism. M is pointless. Proof. If M is not pointless then the identity morphism lies in both E and M and has both domain and codomain M. It follows that each of (1) and (2) implies (4). So suppose that M is pointless. Then (2) must be true, for if m : L → M is an M-morphism then any element a ∈ max L would map to an element m(a) < ⊤ in M, and since m is skinny, we would have πM ◦ m(a) = ⊤. This cannot be the case, since the pointless nucleus πM acts as the identity on M. And (1) must also be true, for if m : M → L is a frame homomorphism then any element a ∈ max L would produce the element m∗ (a) ∈ max M, contrary to assumption. If (3) holds then M is isomorphic to πM and is therefore pointless. And if M is pointless and n : M → N is an E-morphism then n, which is surjective W by definition, must also be one-one because Θn 6 max M Φb = ⊥con M = 1M .  Lemma 4.3.2. plFs is closed under the formation of M-sources. m i Mi )I in Fs such that all the Mi ’s are Proof. Consider a source S ≡ (L −→ pointless. If a is a maximal element of L then because S ∈ M there must be an index i ∈ I for which mi (a) < ⊤. Since mi is skinny, there must be a maximal c ∈ L which lies above m(a), contrary to our assumption that Mi is pointless. We conclude that no maximal element can exist in L.  Theorem 4.3.3. plFs is E-reflective in Fs. Proof. This is an instance of [1, 16.1].  Proposition 4.3.4. An Fs-homomorphism m : L → M is isomorphic to πL if and only if m ∈ E and M is pointless. POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 25 Proof. If M is pointless then m factors through the pointless reflector πL, say m = k ◦ πL . m L → M l ← → πL ← → ← ← → k πL On the other hand, we claim that πL factors through m by Lemma 2.2.4. For if m(a) = ⊤ for some a ∈ L then since m ∈ E we have _ _ { Φc : c ∈ max L and m(c) = ⊤ } ⊆ (a, ⊤) ∈ Φc max L = { (a1 , a2 ) : πL (a1 ) = πL (a2 ) } . (The final equality is a consequence of Proposition 3.1.4.) We conclude that πL (a) = πL (⊤) = ⊤, so that Lemma 2.2.4 provides a homomorphism l : M → πL such that l ◦ m = πL. Since both m and πL are surjective, both k and l are unique with respect to their properties and are therefore inverse isomorphisms.  Corollary 4.3.5. The top arrow in the diagram of Proposition 3.5.4 is the pointless reflector of Ce . Proof. We leave to the reader the routine verification that the arrow Ce → πL is an E-morphism.  4.4. A second look at the spatial reflection. The (E, M)-factorization structure also allows us to refine our understanding of the sF-reflection in F, for it manifests in the form of an M-reflection in Fs. Definition (sFs). We denote by sFs the full subcategory of Fs comprised of the spatial frames. Proposition 4.4.1. The category sFs is M-reflective in Fs. In fact, the sFreflector σL : L → σL of a frame L is an M-surjection which functions as its sFs-reflector. Proof. Because the nucleus σL fixes maximal elements it is clearly skinny and an M-morphism. To check the reflective property, consider a skinny frame homomorphism m with spatial codomain M, and let m = l ◦ σL be its factorization given by Proposition 3.1.4. Then l must be skinny by Corollary 4.1.2(4).  26 R. N. BALL Proposition 4.4.2. A frame homomorphism n : L → N is isomorphic to σL if and only if n is an M-surjection and N is spatial. Proof. If N is spatial then it factors through the spatial reflector σL , say n = k ◦ σL . n L → N → σL ← ← ← → k σL The map k is surjective because n is; to demonstrate that it is also one-one it is sufficient to establish the claim that the only element a ∈ σL for which k(a) = ⊤ is a = ⊤. For that purpose consider an element a ∈ L such that k(a) = ⊤, and suppose for the sake of argument that k(a) < ⊤. Because σL is spatial there must be an element b ∈ max σL such that b > a, hence k(b) = ⊤. But then c ≡ σL∗ (b) ∈ max L has the feature that n(c) < ⊤ because n ∈ M. In view of the fact that ⊤ > n(c) = k ◦ σL (c) = k ◦ σl ◦ σL∗ (b) = k(b) = ⊤, we have the contradiction which proves the claim. 5. T HE  POINTLESS AND SPATIAL REFLECTIONS TOGETHER In this section we address the interactions between the pointless and spatial parts of a frame. The two parts are bound together by a function which will play an important role in what follows. 5.1. The ligature binding the pointless and spatial parts of a frame. Definition (ligature λL : πL → πσL). It is an immediate consequence of Theorem 4.3.3 that for any frame L there is a unique frame surjection λL such that λL ◦ πL = πσL ◦ σL . σL L → σL ← ← ← ← πL → πσL → πL → πσL λL We refer to this map as the pointless ligature of L. Within a given frame L reside the three sublocales πL, σL, and πσL. Lemma 5.1.1 provides a visualization of the actions of the three corresponding nuclei on an arbitrary frame element. This picture brings to light isomorphisms between certain subintervals of the frame. POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 27 Lemma 5.1.1. For an element a of a frame L, let b ≡ λL ◦πL(a) = πσL ◦σL (a). b ← ← ← πL (a) ← ← σL (a) ← ← ← a Then [a, πL(a)]L ≈ [σL (a), b]L and [a, σL(a)]L ≈ [πL (a), b]L. Proof. In view of the fact that πL (a) ∧ σL (a) = a by Lemma 3.5.1, this is an application of Lemma 2.4.1(1).  5.2. The subdirect (E, M)-product decomposition of a frame. Definition (subdirect (E, M)-product). A subdirect (E, M)-product of frames E and M is a subframe L of the product frame E × M such that the projection map L → E is an E-surjection and the projection map L → M is an M-surjection. If E happens to be pointless then the projection L → E is (isomorphic to) the pointless reflection of L by Proposition 4.3.4. If M happens to be spatial then the projection m : L → M is (isomorphic to) the spatial reflection of L by Proposition 4.4.2. Our first task is to show that a subdirect (E, M)-product of a pointless frame E with an arbitrary frame M has a normal form. We produce this normal form by first identifying a subset L ′ of the completely regular frame E × M which is closed under the frame operations, and then passing to its completely regular coreflection, i.e., to the largest completely regular frame L contained in L ′ . This coreflection is a subdirect (E, M)-product of E and M as long as the projections are surjective. The latter condition is not generally met, but is met in the cases of interest (Theorem 5.2.2). Proposition 5.2.1. Let E be a pointless frame and M be a spatial frame, and let l : E → πM be a frame surjection. Then L ′ ≡ { (a, b) ∈ E × M : l(a) = πM (b) } is a naked subframe of E × M, and if the completely regular coreflection L ⊆ L ′ has surjective projections then (1) max L = { (⊤, a) : a ∈ max M }, (2) the projection m : L → M is an M-morphism, 28 R. N. BALL (3) the projection e : L → E is an E-morphism, and (4) l ◦ e = πM ◦ m. In short, L is a subdirect (E, M)-product of E and M with ligature (isomorphic to) l. Proof. Let us show that L ′ is closed under the frame operations of E × M. We can see that L ′ contains both ⊥E×M and ⊤E×M . To show that L ′ is closed under binary meets, simply note that if (ai, bi ) ∈ L ′ then l(a1 ∧ a2 ) = l(a1 ) ∧ l(a2 ) = πM (b1 ) ∧ πM (b2 ) = πM (b1 ∧ b2 ). To show that L ′ is closed under arbitrary joins, consider a subset {(ai , bi)}I ⊆ W W W W L ′ . Then l ( I ai ) = I l(ai ) = I πL (bi) = πL ( i bi ). Now suppose that the projections e and m of the completely regular coreflection L ⊆ L ′ are surjective; we get that l ◦ e = πM ◦ m by construction. πM → → e ← E → M ← m ← ← L l → πM (1) It is obvious that the maximal elements of L ′ are of the form (⊤, a), a ∈ max M. Now consider a maximal element (a, b) of L. We claim that a = ⊤, for if not then since max E = ∅ there exists an element a ′ ∈ E such that ⊤ > a ′ > a, and since e is surjective there exists an element c ∈ M for which (a ′ , c) ∈ L. Because (a ′ , c)  (a, b) ∈ max L, it follows that (⊤, ⊤) = (a ′ , c) ∨ (a, b) = (a ′ , b ∨ c), hence a ′ = ⊤, a contradiction which proves the claim. Thus our maximal element of L has the form (⊤, b). Our second claim is that b ∈ max M. For M is spatial and b < ⊤, so there exists an element c ∈ max M such that b 6 c. But since m is surjective there must be an element a ∈ E for which (a, c) ∈ L, hence (a, c) ∨ (⊤, b) = (⊤, c) ∈ L, so b = c by maximality. Finally, for every c ∈ max M the surjectivity of m guarantees that m∗ (c) exists in max L, and if m∗ (c) = (a, b) then the preceding argument establishes that a = ⊤ and b = c. The bijective correspondence between max L and max M makes (2) obvious. To verify that e is an E-morphism, note that for (ai, bi) ∈ L, e(a1 , b1 ) = e(a2 , b2 ) ⇐⇒ a1 = a2 =⇒ πM (b1 ) = l(a1 ) = l(a2 ) = πM (b2 ), POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 29 so that with the aid of Lemma 2.3.3 we get Θe 6 { ((a, b1 ), (a, b2)) : a ∈ E and (b1 , b2 ) ∈ ΘπM } ! _ _ Φc = m−1 (Φc ) = m−1 (ΘπM ) 6 m−1 = _ max M
max M Θm ∨ Φm∗ (c) = Θm ∨ max M In view of the facts that _ Φm∗ (c) . max M Θe ∧ Θm = ⊥con L = 1L and max L = {m∗ (c)}max M , W W we conclude that Θe 6 max M Φm∗ (c) = max L Φa .  Notation (E ×l S). For a pointless frame E, a spatial frame M, a frame surjection l : E → πM, we denote the frame L of Proposition 5.2.1 by E ×l M. Theorem 5.2.2. Every frame is a subdirect (E, M)-product of its pointless and spatial parts. In detail, for any frame L the product morphism πL × σL : L → πL × σL factors through the inclusion πL ×λL σL → πL × σL, and the initial factor is one-one. σL → πL ×λL σL ≡ τL πL ← ← → → πL → σL τL → ← ← ← L → ← πL × σL We denote this initial factor by τL : L → πL ×λL σL ≡ τL. Proof. The factorization is a consequence of the fact that λL ◦ πL = πσL ◦ σL . The one-oneness of the product map is a consequence of Lemma 3.5.1, from which the one-oneness of its initial factor τL follows. The image τL (L) is contained in the subset L ′ of Proposition 5.2.1, and since that image is completely regular, it is contained in the largest completely regular subset of L ′ , namely πL ×λL σL.  5.3. The fat reflection in Fs. 30 R. N. BALL Definition (fat frames). Let us agree to call a frame fat if the map τL : L → τL of Theorem 5.2.2 is surjective. We denote the full subcategory of Fs comprised of the fat frames by fFs. Theorem 5.3.1. fFs is bireflective in Fs, and a reflector for the frame L is the map τL : L → τL of Theorem 5.2.2. Proof. Provided it is skinny, a test morphism m : L → M factors through both the pointless and spatial reflectors of L and M, and therefore engenders unique maps from each object of the diagram of Theorem 5.2.2 for L to the corresponding object of the diagram for M. Since M is fat, it is isomorphic to τM, so we end up with a unique arrow τL → M making both diagrams, and the arrows between them, commute.  We characterize the fat reflector of an atomless frame in Theorem 6.3.3. 5.4. The fat question. Theorem 5.2.2 provides a method for investigating the structure of a frame by analyzing the interplay between its spatial and pointless parts, and this article can be regarded as first steps towards such an analysis. Although our results are preliminary rather than conclusive, we believe that the questions they raise are important. Chief among them is Question 5.4.1. Question 5.4.1. What are the fat frames? Under what circumstances is the representation τL : L → τL surjective, i.e., when is τL (L) = πL×λL σL? Proposition 6.2.4 shows an atomless frame with finite spatial part is always fat, but the general question is wide open. 6. ATTACHING POINTS TO A POINTLESS FRAME In this section we construct various frames L having a given pointless frame E as their pointless part. The task has already been done if E is empty, since the scattered frames, and only the scattered frames, have the empty frame as their pointless part. So we assume throughout this section that E is a given frame which is nonempty and pointless, and therefore also atomless. 6.1. Attaching finitely many points to a pointless frame. In this subsection we sprinkle finitely many points into the pointless frame E at whim. That is, we show that it is possible to locate the points anywhere in E we please, subject only to the necessity of using an independent family of regular filters to do so. POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 31 Notation (2I , J). We denote the two-element frame by 2, and for an index set I we denote the product of I copies of 2, i.e., the frame of functions I → 2, by 2I . Of course, every element of 2I is the characteristic function of a unique subset J ⊆ I; we denote this function by J, the function itself being defined by the rule  ⊤ if i ∈ J J(i) = , i ∈ I. ⊥ if i ∈ /J Proposition 6.1.1. Let W be a nonempty finite independent family of regular filters on E. For each a ∈ E, let Wa ≡ { w ∈ W : a ∈ w }. Then 
LW ≡ a, Y ∈ E × 2W : Y ⊆ Wa is a completely regular subframe of E × 2W with the following features. (1) LW is atomless.
(2) max LW = ⊤, W \ {w} . W (3) The spatial reflection of LW is
 σLW : LW → σLW = ⊤, Y : Y ⊆ W


= a, Y 7→ ⊤, Y , a ∈ E, Y ⊆ W. The reflection frame σLW is isomorphic to 2W , and the reflector is (isomorphic to) the projection onto the second factor:. (4) The pointless reflector of LW is 
πLW : LW → πLW = a, Wa E


= a, Y 7→ a, Wa , a ∈ E, Y ⊆ W. The reflection frame πLW is isomorphic to E, and the reflector is (isomorphic to) the projection onto the first factor. (5) The functions πσLW and λLW have empty codomains. (6) In terms of Proposition 5.2.1, LW is (isomorphic to) E ×λLW 2W .
Proof. To show that LW is a subframe of E × 2W , consider ai , Yi ∈ LW .


Then a1 , Y1 ∧ a2 , Y2 = a1 ∧ a2 , Y1 ∩ Y2 and a1 ∧ a2 ∈ ∩(Y1 ∩ Y2 ) hence


S a1 , Y1 ∧ a2 , Y2 ∈ LW . Likewise if ai, Yi I ⊆ LW and Y ≡ I Yi then


W W W W I ai ∈ ∩Y, hence I ai , Y and I ai , Yi ∈ LW . The proof I ai , Yi = that LW is completely regular is completed by the Lemmas 6.1.2, 6.1.3, and 6.1.4, in which the symbols retain their meaning in the theorem statement. 32 R. N. BALL
(1) Suppose for the sake of argument that a, Y is an atom of LW . If Y = ∅ then a would have to be an atom of E, which is ruled out by hypothesis. If Y contains two distinct elements yi ∈ Y then we would have


⊥ < a, Y \ {y1 } 6= a, Y \ {y2 } < a, Y , contrary to assumption. So Y must be a singleton, say Y = {y}, and a must lie in y. But Lemma 6.2.3 assures the existence of an
element
b ∈ y such that b < a, so that we have the contradiction ⊥ < b, Y < a, Y . We conclude no atom exists in LW . (2) Since E is pointless, it is clear
that the only predecessors of ⊤E×2W =
⊤, W are of the form ⊤, W \ {w} for elements w ∈ W. (3) Thus the
maximal elements above a typical member a, Y ∈ LW are those of the

form ⊤, W \ {w} for w ∈ W \ Y, and their meet is ⊤, Y . (4) And with Lemma
2.4.3(3) in mind, it is clear
that the successors of a typical element a, Y are of the form a, Y ∪ {w} for elements w ∈ W \ Y, and the join of
 these successors is then a, Wa . Notation (partition W = Y ⊕ Z). We write W = Y ⊕ Z to indicate that subsets Y, Z ⊆ W partition W, i.e., Y ∪ Z = W and Y ∩ Z = ∅. Lemma 6.1.2. For any nontrivial partition W = Y ⊕ Z and for any z ∈ Z, _ b∗ = ⊤. z ′ ≡ { b ∈ z : b∗ ∈ ∩Y } 6= ∅ and z′ Proof. Since z is regular, it is enough to show that z ′ is nonempty. But since W is independent, for each y ∈ Y there exists an element by ∈ z such that V b∗y ∈ y, yielding Y by ∈ z ′ .  Lemma 6.1.3. Let W = Y ⊕ Z be a nontrivial partition, let t : Z → E be a function such that t(z) ∈ z ′ for
all z ∈ Z,
and let a, c ∈ ∩Y be
such that
c ≺≺ V a 6 Z t(z)∗ . Then both c, Y and a, Y lie in LW , and c, Y ≺≺ a, Y . Proof. It is sufficient
to show
that under the weaker assumption a ≺ c we can conclude that c, Y ≺ a, Y . For in the presence of this weaker result, the stronger hypothesis that c ≺≺ a posits the existence of a witnessing family {aq }Q such that c 6 aq 6 a for all q ∈ Q, whereupon the weaker conclusion
 produces a witnessing family aq , Y Q establishing the stronger conclusion

c, Y ≺≺ a, Y .
W Therefore assume that c ≺ a, and consider the element c∗ ∨ Z t(z), Z , element an element which
lies in
LW by construction. We claim that this
∗ W t(z) = witnesses c, Y ≺≺ a, Y . The claim relies on the fact that Z POINTLESS PARTS OF COMPLETELY REGULAR FRAMES V Z W t(z)∗ , so that c ∧ Z t(z) = ⊥. Therefore  _

t(z), Z = ⊥E , ∅ = ⊥LW and c, Y ∧ c∗ ∨ Z 
 ∗ _
a, Y ∨ c ∨ t(z), Z = ⊤E , W = ⊤LW . Z 33  Lemma 6.1.4. LW is completely regular.
Proof. Given the element a, Y ∈ LW , put Z ≡ W \ Y. Let T be the family of all functions t : Z → E such that t(z) ∈ z ′ for all z ∈ Z. For each t ∈ T let ^ C(a, t) ≡ c ∈ ∩Y : c ≺≺ a ∧ t(z)∗ . Z W Note that since a ∈ ∩Y and each filter y ∈ Y is round, C(a, t) = a ∧ V ∗ Z t(z) . Then ! ! ! ^ __ _ _ _
a ∧ t(z)∗ , Y = C(a, t), Y = c, Y = T T C(a,t) a∧ _^ T t(z)∗ , Y Z ! Z T = a∧ ^_ Z t(z)∗ , Y T !
= a, Y .  Lemma 6.1.4 completes the proof of Proposition 6.1.1. Proposition 6.2.4 then completes the circle of ideas by showing that LW is characterized by its properties. This requires the consideration of the spatial support of a frame. 6.2. The spatial support of a frame. Notation (xa , ya , spatial support). In a frame L with pointless part E, we denote the ideals in L and E generated by an element a ∈ max L by   xa ≡ b ∈ L : b  a and ya ≡ b ∈ E : b  a . The spatial support of L is {ya }max L . Lemma 6.2.1. In a frame L with pointless part E and maximal element a, ya = πL (xa ), xa = π−1 L (ya ), and ya = xa ∩ E. Lemma 6.2.2. Let L be an atomless frame with pointless part E. (1) If x is a proper regular filter on L then πL(x) is a proper regular filter on E. (2) If x is a maximal proper round filter on L then πL (x) is a proper regular filter on E. 34 R. N. BALL (3) If A is a subset of max L then {πL (xa )}A is an independent family of proper regular filters on E. Proof. (1) We can identify E with πL, and think of πL as a dense surjection from L onto E. One consequence is that πL∗ (⊥) = ⊥, so that πL(x) is proper because x is proper. Another consequence is that the skeleton map δL factors through πL so that the skeleton itself is contained in πL. That implies that for an element a ∈ πL, its pseudocomplements in L and πL coincide. Thus for a regular filter x on L, ! _ _ _ _ πL (x)∗ . ⊤= b∗ =⇒ ⊤ = πL b = πL (b∗ ) = x x x x (2) If x is a maximal proper round filter on L which is not regular then it is of the form xa for some a ∈ max L by Lemma 2.5.1(5), in which case πL (x) is regular by Lemma 2.5.3(4). (3) If a1 and a2 are distinct maximal elements of L then xa1 and xa2 are distinct maximal proper round filters by Lemma 2.5.1(6), and therefore contain disjoint elements bi  ai by Lemma 2.5.1(4). Since πL is a dense surjection, πL(b1 ) and πL (b2 ) are disjoint elements of πL (xa1 ) and πL (xa2 ), respectively.  Lemma 6.2.3. A proper regular filter on any frame cannot contain a least element. Proof. If a proper regular filter x on a frame L contained a least element a then a would have to be completely below itself and therefore be compleW mented. But in that case x b∗ = a∗ < ⊤, contrary to hypothesis.  Proposition 6.2.4. An atomless frame M having pointless part E and finite spatial support W is isomorphic to LW . Explicitly, the map

a ∈ M, k : M → LW ≡ a 7→ πM (a), Wa , is an E-isomorphism, and in light of of Proposition 6.1.1(6), is (isomorphic to) the map τM from Theorem 5.2.2. In particular, an atomless frame with finite spatial part is fat. Proof. The reader will have no difficulty verifying that k preserves binary meets and therefore preserves and reflects order. The proof that k is a frame isomorphism is completed by showing that k is bijective. That k is one-one follows from Lemma 3.5.1, for if k(a1 ) = k(a2 ) then πM (a1 ) = πM (a2 ), and POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 35 σM (a1 ) = σM (a2 ) because a1 and a2 lie below the same maximal elements by Proposition 6.1.1(2).
To check that k is surjective, consider an element b, Z ∈ LW . By definiT tion of LW , b ∈ E and b ∈ Z ya . Let A ≡ { a ∈ max M : b ∈ ya ∈ / Z},
and put b ′ ≡ b ∧ A. We aim to show k(b ′ ) = b, Z . Surely πM (b ′ ) = V πM (b) ∧ A πM (a) = πM (b) = b; what remains to be shown is that Z = { ya : b ′ ∈ ya }. But this is clear, for by construction it is the case that for any a ∈ max L, b ′ 6 a if and only if b 6 a or a ∈ A. That is, b ′ ∈ ya if and only if b ∈ ya and a ∈ / A, i.e., if and only if ya ∈ Z. Finally, k is an E-morphism because for bi ∈ M, k(b1 ) = k(b2 ) if and only if πM (b1 ) = πM (b2 ) .  V Proposition 6.2.5 is preparation for Subsection 6.3 Proposition 6.2.5. Let M be an atomless frame with pointless part E, let X be W a nonempty finite subset of its spatial support W, and let Ξ ≡ { Φa : ya ∈ / X }. Then the quotient map kX : M → M/Ξ is an E-morphism with codomain (isomorphic to) LX . Proof. The map kX is an E-morphism by construction, and its adjoint kX∗ provides a bijection from max(M/Ξ) onto X. The result follows from Proposition 6.2.4.  6.3. Attaching infinitely many points to a pointless frame. The next task is to analyze atomless frames with infinite spatial support. Lemma 6.3.1. Let X and Y be nonempty finite independent families of regular filters on E, let r : Y → X be a function such that y ⊇ r(y) for all y ∈ Y, and let
Z ⊆ X, r−1 : 2X → 2Y = Z 7→ r−1 (Z) , be the frame map induced by r. Then the map


lX a, Z 7→ a, r−1 (Z) , Y : LX → LY ≡ a ∈ E, Z ⊆ X, is a skinny frame injection.
Proof. A maximal element of LX , which has the form ⊤, X \ {x} for some
x ∈ X, maps to ⊤, Y \ r−1 (x) ≡ b. Enumerate r−1 (x) as {yi }i6n . If n = 0
then b = ⊤, Y = ⊤LY . If n > 0 then for each i 6 n,
bi ≡ ⊤, Y \ { yj : j 6= i } 36 R. N. BALL W is a successor of b, and i6n bi = ⊤. In either case πLY (b) = ⊤, meaning that lX  Y is skinny by Lemma 4.1.1(3). Notation (Y ⊆ω W). Let W be a family of filters on a frame L. We write Y ⊆ω W to indicate that Y is a finite subset of W. Definition (plenary family of regular filters on E). We shall call a family W of regular filters on E plenary if it is nonempty and independent, and for all ⊥ < a ∈ E there exists an element w ∈ W such that a ∈ w, i.e., Wa 6= ∅. Proposition 6.3.2. Let W be a plenary family of regular filters on E, and for subsets Y ⊆ X ⊆ω W, let lX Y : LX → LY be the surjection of Lemma 6.3.1 arising from the inclusion Y → X. The nF-pullback (limit) lW X ′ −→ LX : X ⊆ω W LW is (isomorphic to) lW X : LW of the diagram lX Y LY : Y ⊆ X ⊆ω W LX −→
a, Z ∈ E × 2W : Z ⊆ Wa with projections


→ LX = a, Z 7→ a, Z ∩ X , Z ⊆ W, a ∈ E.  ′ Then the completely regular coreflection LW of LW is the Fs-pullback of the diagram. If the projections are surjective then LW is atomless and 
′ max LW = max LW = ⊤, W \ {w} : w ∈ W . If the projections are E-morphisms then the pointless reflector of LW is 
πLW : LW → πLW = a, Wa : a ∈ E



= a, Z 7→ a, Wa , a, Z ∈ LW . The reflection frame πLW is isomorphic to E, and the reflector is (isomorphic to) the projection onto the first factor. Proof. A moment’s reflection is all that is necessary to check the first paragraph, so we begin by verifying that LW is atomless when the projections
are surjective. Were it to exist, an atom of LW would have the form a, Z for

′ . It follows that a > ⊥ in E, for otherwise ⊥LW = ⊥E , ∅ < a, Z ∈ LW

Z ⊆ Wa = W⊥ = ∅ so a, Z = ⊥, ∅ , contrary to assumption. Since W is plenary, there exists an element w ∈ Wa , so that
for any subset
X ⊆ω W conW taining w, the surjection lX would take a, Z to a, Z ∩ X ∈ LX . But
since LX is atomless by Proposition 6.1.1(1), there exists an element b, Y ∈ LX

such that ⊥LX < b, Y < a, Z ∩ X , and since lW X is surjective, there exists POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 37



b, V = b, V ∩ X = b, Y . It follows an element b, V ∈ LW such that lW X



that b, V ∩ Z = b, V ∧ a, Z lies in LW and satisfies ⊥LW < b, V ∩ Z <
a, Z , contrary to assumption. We conclude that LW is atomless.
A maximal element a, Z ∈ LW has the feature that each of its projections

lW X a, Z = a, Z ∩ X , X ⊆ω W, must be either the top element of LX or a maximal element of LX , and
the second for
case must
occur. To reiterate,
all X ⊆ω W either a, Z ∩ X = ⊤, X or a, Z ∩ X = ⊤, X \ {x} for some x ∈ X. In any case a = ⊤, and the element x in the second case is unique. For if x1 and x2 are distinct elements
of X for which there exist subsets Xi ⊆ω W
W such that xi ∈ Xi and lXi ⊤, Z = ⊤, Xi \ {xi } then X ≡ X1 ∪ X2 ⊆ω W


and lW X ⊤, Z = ⊤, X \ {x1 , x2 } , a clear contradiction since ⊤, X \ {x1 , x2 } is not maximal in LX . In short, a maximal element of LW must be a maximal ′ element of LW . Furthermore, the surjectivity of the projections forces every ′ maximal element of LW to appear as a maximal element of LW . If the projections are E-morphisms then the Fs-pullback is a source comπL X prised of E-morphisms. Likewise the sink LX −−→ E : X ⊆ω W is comprised of E-morphisms, and since the composition of E-morphisms is an E-morphism ([1, 5.1.8(2)]), the composition of this source with this sink is a single E-morphism LW → E. This morphism must be the pointless reflector of LW by Proposition 4.3.4.  Notation (LW for arbitrary W). For a plenary family W of regular filters on a pointless frame E, we denote the frame constructed in Proposition 6.3.2 by LW . We summarize. Theorem 6.3.3. An atomless frame M with pointless part E and spatial support W is isomorphic to a subframe of LW with surjective projections. In detail, the source { kX : M → LX : X ⊆ω W } provided by Proposition 6.2.5 factors through the pullback LW of Proposition 6.3.2, and this embedding is the fat reflector of M (Theorem 5.3.1). Theorem 6.3.3 focuses attention on the fundamental Question 6.3.4. Question 6.3.4. In a given pointless frame E, which plenary families of regular filters serve as the spatial support of atomless frames with pointless part E? 38 R. N. BALL 7. T HE SPATIAL SUPPORT OF A FRAME In this section we focus on the interaction between a frame and its spatial support. In Subsection 7.1 we characterize the spatiality and compactness of the frame in terms of its spatial support, and in Subsection ?? we look into the situation that arises when all of the filters in the spatial support of the frame are maximal proper round filters on its pointless part. 7.1. The spatial support determines spatiality and compactness. We show in Proposition 7.1.2 that a necessary condition for an atomless frame to be spatial is that it must contain enough regular filters in its spatial support to distinguish the elements of its pointless part. This requires the preparatory Lemma 7.1.1, in connection with which recall the inductive definition of πL given in Section 2. Lemma 7.1.1. For an element b of a frame L, define A ′ (b) ≡ { a ∈ max L : a > b and a → b > b } , and for ordinal numbers β define Aβ (b) ≡ {b} if β = 0, Aβ (b) ≡ Aα (b) ∪ A ′ (πα L (b)) if β = α + 1, [ Aβ (b) ≡ Aα (b) if β is a limit ordinal, and α<β A(b) ≡ Aβ (b) for some (any) β such that Aβ (b) = Aβ+1 (b). Then b = πL (b) ∧ V A(b). Proof. We first claim that b = πL′ (b) ∧ V A ′ (b) for any b ∈ L. For   ^  ^  _ A ′ (b) ∧ πL′ (b) = A ′ (b) ∧ b ∨ (a → b)  = b∧ ^ =b∨ ′  A (b) ∨ _ ^ A ′ (b) _ ^ A ′ (b) ′  ′  A ′ (b) A (b) ∧ (a → b)   A (b) ∧ (a → b) = b. V β We then use induction to show that b = πβ (b) ∧ A (b) for all β. The L assertion clearly holds for β = 0, so assume it holds for all ordinals α < β. POINTLESS PARTS OF COMPLETELY REGULAR FRAMES If β = α + 1 then 39 ^ Aβ (b) = πα+1 (b) ∧ Aα+1 (b) = L ^ (Aα (b) ∪ A ′ (πα πL′ ◦ πα L (b) ∧ L (b))) =   ^ ^ πL′ ◦ πα A ′ (πα Aα (b) = L (b) ∧ L (b)) ∧ ^ πα (b) ∧ Aα (b) = b. L πβ L (b) ∧ ^ (The penultimate equality holds by the claim and the ultimate equality by the induction hypothesis.) If β is a limit ordinal then we get ^ _ ^[ β α (b) ∧ A (b) = π (b) ∧ Aγ (b) = b 6 πβ L L _ α<β πα L (b) ∧ ^[ γ A (b) γ<β α<β ! 6 γ<β _ πα L (b) ∧ α<β ^ α  A (b) = b.  Proposition 7.1.2. The following are equivalent for a frame L with pointless part E.   \ \ ∀b ∈ L ↑b↑L = xa = xa  , (1) ba∈max L (2) ∀b ∈ L b= ^ a ! , b∈xa i.e., L is spatial, b6a∈max L (3) ∀b ∈ E b= ^ a ! . b6a∈max L When these conditions obtain then  ∀b ∈ E ↑b↑E = \  ya  . ba∈max L Proof. (2) is equivalentx to (3),x for if (3) holds then (2) follows from Lemma 7.1.1 because A(b) ∪ πL (b)max L ⊆ ↑b↑max L . Assume (1) and suppose for V the sake of argument that (2) fails at b ∈ L, i.e., b < b ′ ≡ ↑b↑max L . Then b∈ / ↑b ′ ↑L , so that by (1) there must be some element a ∈ max L for which b∈ / xa ∋ b ′ , i.e., b ′  a > b, contrary to assumption. Conversely, assume T (2) to prove (1). Clearly we have ↑b↑L ⊆ { xa : b ∈ xa , a ∈ max L } for any 40 R. N. BALL b ∈ L. If b  b ′ ∈ L then by (2) there exists a maximal element a such that b  a > b ′ , i.e., b ′ ∈ / xa ∋ b. If L satisfies the numbered conditions then for any b ∈ E we have \ ↑b↑E = E ∩ ↑b↑L = E ∩ { xa : b ∈ xa , a ∈ max L } \ { E ∩ xa : b ∈ xa , a ∈ max L } = \ { ya : b ∈ ya , a ∈ max L } =  Question 7.1.3. Are the numbered conditions of Proposition 7.1.2 equivalent to the condition given in the last sentence? Proposition 7.1.4. Let L be an atomless spatial frame with pointless part E and spatial support W. Then L is compact if and only if W is maximal among independent families of regular filters on E. Proof. If W is not maximal then there exists a regular filter z on E such that W ∪{z} is independent, i.e., for each w ∈ W there exist an element b ∈ z such W that b∗ ∈ w. Then z b∗ = ⊤L , for otherwise the spatiality of L would imply the existence of some a ∈ max L for which b∗ 6 a for all b ∈ z, i.e., b∗ ∈ / ya ′ for all b ∈ z, contrary to assumption. But no finite subset z ⊆ω z has the W V feature that z ′ b∗ = ⊤, for that would imply that the element b0 ≡ z ′ ∈ z had the feature that b∗0 = ⊤, an impossibility. The point here is that L is not compact. Now suppose that W is maximal, and suppose for the sake of argument that C is a cover of L without a finite subcover. By replacing C with the the ideal it generates, we may assume that C is a proper ideal. By replacing C with { b : ∃c ∈ C (b ≺≺ c) }, we may assume that C is proper and round. C is therefore contained in a maximal proper round ideal C ′ , and x ≡ { c∗ : c ∈ C ′ } is a maximal proper round filter on L by Lemma 2.5.1(10), so that y ≡ πL (x) is a regular filter on E by Lemma 6.2.2. By virtue of its maximality, W must contain at least one filter w with which y has the finite intersection property, which is to say that a ∧ b > ⊥ for each a ∈ w and b ∈ y. Since W is the spatial support of L, there exists a unique  element a  ∈ max L for which w = ya = b ∈ E : b  a . We claim that ′ x = x ≡ b ∈ L : b  a . For if x differs from x ′ then x and x ′ would contain disjoint elements by Lemma 2.5.1(4), an eventuality which is ruled out by the observation that πL (x ′ ) = w and πL (x) = y, and w and y have W the finite intersection property. The important point here is that a = x b∗ by Lemma 2.5.1(5). POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 41 The argument is concluded by noticing that for each c ∈ C we have c∗ ∈ x, W W hence c∗∗ 6 x b∗ = a, from which follows C 6 a. This contradicts our assumption that C covered L. We conclude that L is compact.  Corollary 7.1.5. An atomless frame L whose spatial support is maximal among V independent families of round filters has the feature that ↑b↑L = b∈xa xa for all b ∈ L. Proof. A compact frame is spatial. 8. W HEN  THE SPATIAL SUPPORT CONSISTS OF MAXIMAL PROPER ROUND FILTERS In this section we show that the condition of complete regularity, which intrudes into our fundamental Proposition 6.3.2, is automatically satisfied when the spatial support consists of maximal proper round filters. The result is Theorem 8.1.4, which requires some preparation. 8.1. Attaching points at maximal proper round filters. Lemma 8.1.1. Let W be a plenary family of maximal proper round filters on E. Then
 ′ LW ≡ a, Z ∈ E × 2W : Z ⊆ Wa is a naked subframe of E × 2W with the following features. ′ (1) LW is atomless.   . ⊤, W \ {w} W 
′ (3) LW contains E ′ ≡ a, Wa E as a subset isomorphic to E in the inher

ited order, and the map E → E ′ = a 7→ a, Wa preserves ⊥, ⊤, and binary meets. The map also preserves the completely below relation in the following sense.
 (4) If {bp }Q is a witnessing family for a1 ≺≺ a2 in E then bp , Wbp Q is

′ . a witnessing family for a1 , Wa1 ≺≺ a2 , Wa2 in E ′ and also in LW

′ In particular, a1 ≺≺ a2 in E implies a1 , Wa1 ≺≺ a2 , Wa2 in LW . ′ (2) max LW = ′ Proof. To show that LW is closed under the frame operations in E ×
2W ,
′ first note LW contains both ⊥E×2W = ⊥, ∅ and ⊤E×2W = ⊤, W . If
that ′ ai , Zi ∈ LW then Z1 ∩ Z2 ⊆ Wa1 ∩ Wa2 = Wa1 ∧a2 , hence


′ a1 , Z1 ∧ a2 , Z2 = a1 ∧ a2 , Z1 ∩ Z2 ∈ LW . 42 R. N. BALL 
S W ′ Likewise if ai , Zi I ⊆ LW and Z ≡ I Zi then I ai ∈ ∩Z, hence

W W ′ I ai , Z ∈ LW . I ai , Zi =
′ (1) Suppose for the sake of argument that a, Z is an atom of LW . If Z = ∅ then a would have to be an atom of E, which is ruled out by hypothesis. If Z contains two distinct elements zi ∈ Z then we would have    
⊥ < a, Z \ {z1 } 6= a, Z \ {z2 } 6 a, Z , contrary to assumption. So Z must be a singleton, say Z = {z}, and a must lie in z. But Lemma 6.2.3 assures the existence of an
element
b ∈ z such that b < a, so that we have the contradiction ⊥ < b, Z < a, Z . (2) Since E is pointless,  it is clearthat the only predecessors of ⊤E×2W =
⊤, W are of the form ⊤, W \ {w} for elements w ∈ W.
(4) It is sufficient to establish the claim that a1 ≺≺ a2 in E implies a1 , Wa1 ≺
′ a2 , Wa2 with witness in LW . So suppose a1 ≺≺ a2 in E with witnessing family {bp }Q, and fix rational numbers p < q. Since b∗p ∨ bq = ⊤, bq ≺≺ a2 with witnessing family {br }q<r , and b∗p ≺≺ a∗1 with witnessing family {b∗r }r<p , it follows from Lemma 2.5.1(9) that Wa2 ∪ Wa∗1 = W. Since Wa∗1 ∩ Wa1 = ∅,


 we have shown that a1 , Wa1 ≺ a2 , Wa2 with witness a∗1 , Wa∗1 . ′ Lemma 8.1.2. Let W and LW be as in Lemma 8.1.1. Then for any w ∈ W,  _
 b∗ , Wb∗ = ⊤, W \ {w} , w the supremum being reckoned in E × 2W . W Proof. The supremum w b∗ is ⊤E because a maximal proper round filter on a pointless frame is regular by Lemma 2.5.1(5). The argument is completed S by showing that w Wb∗ = W \ {w}. For w ∈ / Wb∗ for any b ∈ y since ∗ b ∈ / w. And for any x ∈ W, x 6= w, there exists an element b ∈ w for which ∗ b ∈ x because W is an independent family. The point is that any such x lies S in w Wb∗ .  ′ Lemma 8.1.3. Let W, LW , and E ′ be as in Lemma 8.1.1, and let 
′′ LW ≡ a, Z : Wa \ Z ⊆ω W .
 ′′ ′ Then LW is the smallest subset of LW which contains E ′ ∪ ⊤, W \ {w} W ′′ and is closed under binary meets. Furthermore, each member of LW is the join W ′ (in E × 2 ) of its lower bounds in E . POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 43
Proof. To verify the first statement, observe that a typical element a, Z ∈ ′′ LW can be expressed as

^ a, Wa ∧ ⊤, W \ {w} . Wa \Z To verify the second, use Lemma 8.1.2 to expand this expression.


^
^ _ ∗ b , Wb∗ a, Wa ∧ ⊤, W \ {w} = a, Wa ∧ Wa \Z = _

Wa \Z w a, Wa ∧ b∗w , Wb∗w : bw ∈ w ∈ Wa \ Z .  ′ ′′ Theorem 8.1.4. Let W, LW , LW , and E ′ be as in Lemma 8.1.3, and let L⋆W be ′′ . Then L⋆W has the following the family of all joins (in E×2W ) of elements of LW properties. (1) L⋆W isthe smallest
(completely regular) subframe of E × 2W containing E ′ ∪ ⊤, W \ {w} W . 
(2) max L⋆W = ⊤, W \ {w} W . (3) The pointless part of L⋆W has E as a quotient, in the sense that E ′ is a sublocale of πL⋆W , and if we denote the quotient map by q : πL⋆W → E ′ , then


q ◦ πL⋆W : L⋆W → E ′ = a, Z 7→ a, Wa , a ∈ E, Z ⊆ W. Proof. (1) The fact that L⋆W is completely regular follows from Lemma 8.1.1(3) together with the second sentence of Lemma 8.1.3. E ′ is closed under arbitrary meets, consider a family  (4) To
show that
V V ai , Wai I ⊆ E ′ , and let a0 ≡ I ai in E. We claim that I ai, Wai =

a0 , Wa0 in L⋆W . Since Wa0 ⊆ Wai for all i, it is clear that ao , Wa0 6



ai , Wai for all i. But for any b, Z ∈ L⋆W such that b, Z 6 ai , Wai for all i, it must be the case that b 6 a0 , and hence that Z ⊆ Wb ⊆ Wa0 . ′ ⋆ To
complete the
proof that E is
a sublocale of
LW ,⋆ we shall show
that b, Z → a, Wa = b → a, Wb→a for any b, Z ∈ LW and a, Wa ∈ E ′ .



Surely b → a, Wb→a ∧ b, Z = (b → a) ∧ b, Wb→a ∩ Z 6 a, Wa , for any filter in
Wb→a ∩
Z contains both b
→ a and
b and hence contains a. And if c, Z ′ ∧ b, Z = c ∧ b, Z ′ ∩ Z 6 a, Wa then c 6 b → a hence Z ′ ∩ Z ⊆ Wc∧b ⊆ Wc ⊆ Wb→a .  Proposition 8.1.5. Let W be the family of maximal proper round filters on E. Then the frame L⋆W of Theorem 8.1.4 is (isomorphic to) βE, the compact coreflection (aka Čech-Stone compactification) of E. 44 R. N. BALL Proof. Notice that L⋆W is compact by Proposition 7.1.4. According to the classical construction of Banaschewski and Mulvey ([9], [10]), we may take βE to be the frame of round ideals on E ′ . Therefore it is enough to establish that the maps 



g : L⋆W → βE = b, Z 7→ a, Wa : a, Wa ≺≺ b, Z  _  βE → L⋆W = u 7→ u


W are inverse frame isomorphisms. Clearly g b, Z = b, Z for all b, Z ∈
′′ L⋆W , for b, Z is the join of its lower bounds in LW by construction, each ′′ ′ element of LW is the join of its lower bounds in E by Lemma 8.1.3, and by Lemma 8.1.1, each element of E ′ is the join, in both E ′ and L⋆W , of those elements of E ′ completely below it.
W Consider an element u ∈ βE ′ with u ≡ b, Z . Because u is round, each element of u has an element of u completely above it in E ′ and therefore W
also in L⋆W by Lemma 8.1.1(4), from which it follows that u ⊆ g u . The opposite containment is just as clear, for a round ideal of open subsets of a compact Hausdorff space is precisely the family of open sets completely contained in its union.  Corollary 8.1.6. For an atomless frame L with pointless part E, the cardinality of max L is bounded above by the cardinality of max βE. Example 8.1.7 makes the point that the pointless part of L⋆W (Theorem 8.1.4(4)) may be strictly larger than E. Example 8.1.7. Let E be the regular open algebra of the topology OR of the real numbers, and let W be the family of all maximal round filters on E. Then L⋆W is isomorphic to βE by Proposition 8.1.5, and according to the proof of the proposition, we may take it to be the frame of round ideals on E. Since in this case every ideal is round, L⋆W is (isomorphic to) the topology of the Stone space dual to E, i.e., the topology of the Gleason cover of the real numbers. The pointfree part of L⋆W is the sublocale consisting of those elements which satisfy the condition of Proposition 3.2.4. The ideals which satisfy this condition include the principal ideals, but include other ideals as well. For instance, the reader will have no difficulty in verifying that the ideal of regular open subsets of finite measure is nonprincipal and satisfies the condition. POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 45 Question 8.1.8. Is every pointless frame the pointless part of a compact frame? Proposition 8.1.5, together with Example 8.1.7, motivate the consideration in Subsection 8.3 of compact frames whose pointless parts are C∗ quotients. This requires a brief digression in Subsection 8.2. 8.2. sFs is dually equivalent to Tychs. Here we make the point in Proposition 8.2.1 that the restriction to skinny frame homomorphisms, i.e., the passage from F to Fs, does not invalidate the classical dual equivalence between spatial frames and Tychonoff spaces. Definition (Tychs, max : sFs → Tychsop ). We denote the category of Tychonoff spaces with skinny continuous functions by Tychs. The functor max : sFs → Tychsop associates with each spatial frame L the space with carrier set max L topologized by declaring O max L ≡ {↑a↑max L }L , and associates with each Fs-morphism m : L → M the continuous function
max m : max M → max L = b 7→ m∗ (b) . Proposition 8.2.1 shows that sFs and Tychs are dually equivalent categories. Proposition 8.2.1. The functor max : sFs → Tychsop is an equivalence of categories. Proof. We have already remarked that a continuous function between Tychonoff spaces has a frame counterpart which is skinny if and only if its fibers are scattered, a correspondence which is evidently full and faithful. Of course, every spatial frame is the topology of a Tychonoff space. See [1, 3.33].  8.3. The qP-coreflection in compact atomless frames. Proposition 8.3.1 points out an exotic feature of compact Hausdorff spaces whose pointless parts are C∗ -embedded. Proposition 8.3.1. Let X be a compact Hausdorff space without isolated points whose pointless part is C∗ -embedded, by which we mean that its topology OX ≡ L is isomorphic to βπL. Then the removal of any single point does not change CX, in the sense that for any x ∈ X, CX = C∗ (X \ {x}). 46 R. N. BALL Proof. Denote the open quotient corresponding to X \ {x} by ox : L → Ox . Since πL factors through ox by Proposition 3.1.4, the latter is a C∗ -quotient.  Question 8.3.2. In terms of Proposition 8.3.1, if each quotient ox : L → Ox , x ∈ X, is a C∗ -quotient, does it follow that πL is a C∗ -quotient? The spaces described in Proposition 8.3.1 are reminiscent of P-spaces. A point of a Tychonoff space is called a P-point if every real-valued function on the space is constant on a neighborhood of the point. A P-space is a space whose every point is a P-point. (See [8] and the references therein for a thorough discussion of P-spaces.) Obviously any non-isolated point of a Pspace X can be removed without changing C∗ X. But a compact P-space is finite, whereas we are about to demonstrate that the spaces of Proposition 8.3.1 abound. In fact, they are reflective in compact Hausdorff spaces without isolated points, meaning every such space has a canonical embedding into a space with the properties of Proposition 8.3.1. Definition (qP-frame, qP-space). A qP-frame is a compact frame whose pointless part is a C∗ -quotient. That is, a frame L is a qP-frame if πL : L → πL is isomorphic to βπL : βπL → πL.3 A qP space is a Tychonoff space whose topology is a qP-frame. Lemma 8.3.3. A compact frame with a pointless C∗ -quotient is a qP-frame. Otherwise put, a qP-frame is a frame of the form βM for some pointless frame M. Proof. If m : L → M is a C∗ -quotient map with L compact and M pointless then, because it must factor through πL , it follows that πL is a C∗ -quotient isomorphic to βπL .  Corollary 8.3.4. A qP-frame is atomless. Proof. Because a qP-frame L is of the form βM for a pointless frame M, the coreflector βM : βM → M is surjective and therefore takes atoms to atoms, and because M is atomless by Lemma 2.4.3(2), it follows that L is atomless.  Definition (kalFs, qPFs). We denote by kalFs and by qPFs the full subcategories of Fs comprised, respectively, of the compact atomless frames and of the qF-frames. 3 The term ‘quasi-P frame’ was used in [13] with another meaning. POINTLESS PARTS OF COMPLETELY REGULAR FRAMES 47 Theorem 8.3.5. qPFs is coreflective in kalFs; a coreflector for the frame L is the map qL : βπL → L such that πL ◦ qL = βπL . Proof. Given a test kalFs-morphism m with M a qP-frame, apply the pointless reflection to get πm, to which apply the compact coreflection to get βπm, and then insert the maps qL and qM such that πL ◦ qL = βπL and πM ◦ qM = βπM . m M → L → ← ← → ← πm βπL ← ← ← ← βπM ← βπM qL → → πL → πM ← qM → πL → πM → βπL βπm Then qM is an isomorphism because πM is a C∗ -quotient map and is therefore isomorphic to βπM (cf. Lemma 8.3.3), so that m = qL ◦ (βπm) ◦ q−1 M is the desired factorization of m.  Corollary 8.3.6. The full subcategory of Tychs comprised of the qP-spaces is bireflective in the full subcategory of Tychs comprised of the compact spaces without isolated points. The author would like to express his gratitude to the CECAT gang for stimulating and encouraging discussions regarding the topics under investigation. These Wednesday afternoon Zoom sessions were organized by Chapman University’s Center of Excellence in Computation, Algebra, and Topology. In particular, the author would like to thank Andrew Moshier for suggesting the proof of Lemma 2.4.5. R EFERENCES [1] J. Adámek, H. Herrlich, and G. Strecker, Abstract and Concrete Categories, The Joy of Cats, http://katmat.math.uni-bremen.de/acc, August, 2004. [2] R. N. Ball, Truncated abelian lattice-ordered groups II: the pointfree (Madden) representation, Topol. Appl. 178 (2014), 56–86. [3] R. N. Ball and A. W. Hager, On the localic Yosida representation of an archimedean lattice ordered group with weak order unit, J. Pure Appl. Algebra 70 (1990), 17–43. [4] R. N. Ball, Pointless real valued functions on a completely regular locale, in progress. [5] R. N. Ball, A. W. Hager, and J. Walters Wayland, From λ-skeletal κ-frames to λ-repletions in W: I The scaffolding of a completely regular frame, in preparation. [6] R. N. Ball, A. W. Hager, and J. Walters Wayland, From λ-skeletal κ-frames to λ-repletions in W: II Continuous convergence on RL, in preparation. 48 R. N. BALL [7] R. N. Ball and A. Pultr, On an aspect of scatteredness in the pointfree setting, Portugal Math. 73 No. 2 (2016), 139–152. [8] R. N. Ball, J. Walters-Wayland, and E. Zenk, The P-frame reflection of a completely regular frame, Topol. Appl. 158 No. 14 (2011), 1778–1794. [9] B. Banaschewski and C. J. Mulvey, Stone-Čech compactification of locales, I, Houston J. Math 6 (1980), 301–312. [10] B. Banaschewski and C. J. Mulvey, Stone-Čech compactification of locales, II, J Pure Appl Algebra 33 (1984), 107–122. [11] F. Dashiell, Shaving points, private communication, April 13, 2022. [12] T. Dube, Some ring theoretic properties of almost P-frames, Algebra Univers. 60 (2009) 145–162. [13] T. Dube and J. Nsonde Nsayi, When certain prime ideals in rings of continuous functions are minimal or maximal, Topol. Apple. 192 (2015), 98–112. [14] P. T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982. [15] J. Madden and H. Vermeer, Epicomplete archimedean ℓ-groups via a localic Wosida representation, J. Pure and Appl. Alg. 68 (1990), 243–252. [16] J. Picado and A. Pultr, Frames and Locales: Topology without Points, Frontiers in Mathematics, Birkhäuser, Springer, Basel, 2012. [17] J. Picado and A. Pultr, Separation in pointfree topology, Birkhäuser, Springer Nature Switzerland AG, 2020. (Ball) D EPARTMENT OF M ATHEMATICS , U NIVERSITY 80208, U.S.A. Email address: [email protected] OF D ENVER , D ENVER , C OLORADO