Intervals in Lattices of κ-Meet-Closed Subsets

Order (2004) 21: 137–153 DOI: 10.1007/s11083-004-3716-2 © Springer 2005 Intervals in Lattices of κ-Meet-Closed Subsets MARCEL ERNÉ Institute of Mathematics, University of Hannover, Germany. e-mail: [email protected] (Received: 29 September 2002; in final form: 22 September 2004) Abstract. We study abstract properties of intervals in the complete lattice of all κ-meet-closed subsets (κ-subsemilattices) of a κ-(meet-)semilattice S, where κ is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closure of A ∪ {x} leaves a closed set). Such closure systems have many pleasant geometric and lattice-theoretical properties; for example, they are always weakly atomic, lower locally Boolean and lower semimodular, and each member has a decomposition into completely join-irreducible
elements. For intervals
of κ-subsemilattices, we describe the covering relation, the coatoms, the -irreducible and the -prime elements in terms of the underlying κ-semilattices. Although such intervals may fail to be lower continuous, they are strongly coatomic if and only if every element has an irredundant (and even a least) join-decomposition. We also characterize those intervals which are Boolean, distributive (equivalently: modular), or semimodular. Mathematics Subject Classifications (2000): Primary: 06A12; Secondary: 06B05, 06A23, 52A01. Key words: (weakly) atomic, (strongly) coatomic, complete lattice, extremally detachable, interval, irreducible, meet-closed, prime, semilattice. 1. Introduction: Prime and Irreducible Elements The study of abstract properties of closure systems whose members are subsemilattices, (convex) sublattices or related substructures of a given lattice or semilattice has a long history (see [1] for one typical reference). In what follows, we shall be concerned with the more general situation of κ-subsemilattices, where the involved closure systems often fail to be algebraic, a lack that makes the theory more complicated. However, many nice results known from “finitary convex geometry” (see Edelman and Jamison [6] and Jamison-Waldner [13]) remain valid for such intervals and even for (not necessarily algebraic) extremally detachable closure systems (see [10]). Given a cardinal number κ, we write X ⊆κ Y if X is a subset of Y with (strictly!) less than κ elements (X = Y not excluded). The cardinal  κ is regular if from X ⊆κ Y and X ⊆κ Y for each X ∈ X it follows that X ⊆κ Y . As usual, the least infinite cardinal is denoted by ω. We also agree that X ⊆∞ Y simply means that X is a subset of Y , i.e. X ⊆ Y . Henceforth, κ always stands for an infinite cardinal or the symbol ∞. 138 MARCEL ERNÉ A κ-(meet-)semilattice is a poset S such that each subset X ⊆κ S has a meet  X or simply by X. In particular, S must have a top elein S, denoted by S  ment S ∅. Thus, an ω-semilattice is a meet-semilattice with a top element, and a κ-semilattice with cardinality less than κ (that is, an ∞-semilattice) is just a complete lattice. Put   κ Y = X : X ⊆κ Y (Y ⊆κ S). S The fixed points of the operator κ are the κ-meet-closed subsets of S, alias κsubsemilattices. They form a closure system Sκ . In the case of a regular cardinal κ, the map κ is the corresponding closure operator. However, regularity is not a serious restriction here, because for irregular κ one could pass to the least regular cardinal κ greater than κ; the closure operator of Sκ is then κ instead of κ . Since κ-subsemilattices are the same as κ-subsemilattices, there will be no loss of generality in assuming that κ is always a regular cardinal or the symbol ∞. If S is a complete lattice then the closure system S∞ is the collection of all meetclosed subsets of S (complete subsemilattices), by our convention about the symbol ∞. Of course, κ-join-semilattices and κ-join-closed subsets are defined dually. Many structural investigations of (semi-)lattices involve various types of “indecomposable” elements. An element m of a κ-meet-semilattice Sis κ-meet-irreduc ible (κ-  -irreducible) if it belongs to each X  ⊆κ S with m = S X, and κ-meetprime (κ- -prime) if for each X ⊆κ S with S X
m, there is an x ∈ X with x
m. The set of all κ- -irreducible elements of S is denoted by Mκ S. In the special case κ = ∞ we  speak of completely meet-irreducible
( -irreducible) or completely meet-prime ( -prime) elements, respectively. (κ-)

 -irreducible and (κ-) -prime elements are defined dually. The ω- - and ω- -primes are the ∨and ∧-primes, respectively, in the usual sense. By a κ-frame we mean a lattice L that is also a κ-join-semilattice and satisfies the distributive law   a∧ B= {a ∧ b : b ∈ B} for all a ∈ L and B ⊆κ L.

Obviously, in a κ-frame the κ- -irreducible elements coincide with the κ- prime elements. By definition, for κ = ∞, a κ-frame is a frame in the usual sense, while an ω-frame is merely a distributive lattice with least
element. Notice that an element j of a complete lattice L is -irreducible if and only if  {a ∈ L : a < j } j∨ = is (the greatest element) smaller than j , and that j is  {a ∈ L : j 
a} j∨ =
-prime if and only if INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS 139 is (the greatest element) not greater than or equal to j . Dually, an element m is  -irreducible if and only if  {b ∈ L : m < b} m∧ =  is (the least element) greater than m, and m is -prime if and only if  m∧ = {b ∈ L : b 
m} is (the least element) not less than or equal to m.
∨ There is a well-known isomorphism j → j between the poset of all -prime elements and that of all -prime elements, with inverse m → m∧ . We shall refer to that isomorphism as the canonical prime isomorphism of L (cf. [7]). By definition, a coatom of a complete lattice L is maximal among the elements distinct from the top element ⊤. We call an element p of L superprime if it has (one of) the following three equivalent properties: (1) p = min{b ∈ L :  b ∨ c = ⊤} for some coatom c, (2) p =
c∧ for some -prime coatom c, (3) p is -prime and p∨ is a coatom. The canonical prime  isomorphism induces a one-to-one correspondence between
superprimes and -prime coatoms. Every superprime element is a maximal prime; the converse is also true if each non-maximal element is dominated by a  -prime.
We say a complete lattice L is -irreducibly generated
or a (join-)decomposition lattice if each element x of L is the join of a set of -irreducibles. Such a set will be referred to as a (join-)decomposition of x. A decomposition of x is said to be irredundant if no proper subset has join x, while a least decomposition is contained in every other decomposition of the same element. (The term “minimal” for the latter type of decompositions, used in the translated Russian literature [14, 15], is a bit misleading, because in the usual order-theoretical terminology, “minimal” would be synonymous with “irredundant”.) Finally, a complete lattice L is said to be superalgebraic or an A-lattice if each
of its elements has a join-decomposition into -prime elements, or equivalently, if L is both a (join-) decomposition lattice and a frame. It is well-known that the dual of a superalgebraic lattice is again superalgebraic, and that the A-lattices are, up to isomorphism, just the Alexandroff topologies, i.e. those topologies for which arbitrary intersections of open sets are open, i.e., the closed sets form a topology, too (see [2] and [8]). 2. Closure Systems of κ-Meet-Closed Subsets In the subsequent investigations, we always consider a fixed κ-semilattice S and a κ-meet-closed subset C of S. In particular, C is a κ-semilattice in its own right. 140 MARCEL ERNÉ Recall that Sκ denotes the closure system of all κ-subsemilattices of S. We are interested in the abstract structural properties of the interval consisting of all κ-meetclosed sets between C and S: C := [C, S] = {A ∈ Sκ : C ⊆ A}. Clearly, C is a closure system, being meet-closed in the closure system Sκ . The corresponding closure operator will be denoted by Ŵ. Thus, ŴY = κ (Y ∪ C) = {a ∧ c : a ∈ κ Y, c ∈ C} for Y ⊆ S, and the binary join in C is given by A ∨C B = {a ∧ b : a ∈ A, b ∈ B} for all A, B ∈ C. More generally, for nonempty B ⊆ C, the join in C is the closure of the union:   B , B= X : X ⊆κ C S and, of course, the join of the empty set is the least element C. If the latter consists of the top element of S only then C coincides with the whole lattice Sκ . Each of the results below applies, in particular, to principal filters in the closure system of all κ-meet-closed subsets of an A-lattice, as discussed in [11]. But in the present context, the specific properties of the least element C and the greatest element S of the interval C, as assumed in [11], are not relevant for our general results. It will be convenient to write A − x for A \ {x} if x ∈ A, A + x for A ∪ {x} if x ∈ / A, and A ∨ x or A(x) for the closure Ŵ(A ∪ {x}). Next we recall two concepts originating from convex geometry (see, for example, [6, 13]) that will be crucial for our investigation of intervals of κ-subsemilattices. Given a closure system C on a set S, an extreme point of a subset A ⊆ S is an element x ∈ A that does not belong to the closure of A − x. If A is a member of C, this is tantamount to saying that A − x is closed. The closure system C is said to be extremally detachable if for each A ∈ C, any point x outside A is an extreme point of A(x) (see [10] and [13]). Henceforth, C denotes an interval [C, S] in the complete lattice Sκ of all κ-meet-closed subsets of a κ-meet-semilattice S. The following fact is easily verified: LEMMA 2.1. C is an extremally  detachable closure system. The extreme points of A ∈ C are precisely the κ- -irreducible elements of A that are not in C. INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS 141 For the first claim, observe that if A ∈ C and x ∈ S \ A then A(x) − x = A ∪ {a ∧ x : a ∈ A, x 
a} is κ-meet-closed, hence a member of C. The characterization of extreme points is clear from the definition of the closure operator associated with C. Now, the whole machinery developed for extremally detachable closure systems (see [6, 10, 13]) applies to the present situation. COROLLARY 2.1. Each A ∈ C has the anti-exchange property A = A(x) = A(y) ⇒ x = y. In other words, x
A y ⇔ A(x) ⊆ A(y) defines a (partial) order on S \ A. A copoint of an element x ∈ S is a maximal member of C not containing x (see Jamison-Waldner [13]). COROLLARY 2.2. Distinct elements of S cannot have common copoints in C.
The following characterization of -irreducible elements in C is obvious:
LEMMA 2.2. The -irreducible elements of C are precisely the sets C(b) = Ŵ{b} = C ∪ {b ∧ c : c ∈ C} (b ∈ S \ C), and C(b) − b is the greatest member of C less than C(b). Each member of C, with exception of the
least one, is a union of such sets C(b). Hence, C (and each subinterval of C) is -irreducibly generated. 3. The Covering Relation and Lower Semimodularity We begin with a basic observation that holds for all closure systems having the anti-exchange property (see [10]), and consequently for each of the intervals C: LEMMA 3.1. For A ∈ C and x ∈ S \ A, the following properties are equivalent: (a) Closedness of A + x (i.e. A + x = A(x)). (b) Minimality of x with respect to the order relation
A . (c) The local exchange property: y ∈ A(x) \ A ⇒ x ∈ A(y). By definition, an element a of a poset is covered by another element b (written a ≺ b) if and only if the interval [a, b] contains exactly these two elements. In closure systems with the anti-exchange property, the covering relation is easily described (see [10]). For easy reference, we note the special case of intervals C in the lattice of κ-subsemilattices and add a short proof. 142 MARCEL ERNÉ LEMMA 3.2. For A, B ⊆ S, the following conditions are equivalent: (a) A ∈ C and B = A + m for some minimal element m of (S \ A,
A ). (b) B ∈ C and A = B − m for some extreme point of B, i.e. m ∈ Mκ B \ C. (c) A, B ∈ C and A ≺ B. Proof. Suppose A ∈ C is a proper subset of B ∈ C, and choose an m in B \ A. Then B ′ = A(m) and A′ = B ′ − m are members of C, and m belongs to Mκ B ′ \ C (see Lemma 2.1). Now, if B covers A then the inclusion chain A ⊆ A′ ⊂ B ′ ⊆ B forces the equalities A = A′ and B = B ′ , hence A = B − m. Thus (c) implies (a) and (b) (use Lemma 3.1). Conversely, if B ∈ C and A = B − m for some m ∈ Mκ B \ C, then A is a member of C, too, and clearly B covers A. This yields the implication (b) ⇒ (c), and (a) ⇒ (c) follows from Lemma 3.1. ✷
We have remarked earlier
that a superprime element is always maximal prime. However, a maximal -prime of an interval C = [C, S] need not be superprime. For example, if S is an infinite binary tree enlarged by a top element ⊤ and an extra (co)atom  a incomparable to all non-minimal elements of the tree, then C = {a, ⊤} is -closed in S, and
for the least element ⊥, the set C(⊥) = {⊥, a, ⊤} is the only (hence maximal) -prime element of [C, S] in S∞ . But C(⊥) is not superprime since C(⊥)∨ = C is not a coatom. Our next result remains valid for arbitrary anti-exchange systems (with extreme  points instead of κ- -irreducible elements). LEMMA 3.3. The map x → S − x induces a bijection between Mκ S \ C and the set of all coatoms of the lattice C, while the map x → C(x) induces a bijection between Mκ S \ C and the set of all superprime elements of C. Proof. By Lemma 3.2, the coatoms  of C are precisely the sets S − x with x ∈ Mκ S \ C, and these are clearly -prime. The canonical prime map establishes  a one-to-one correspondence C(x) → S − x between superprimes and ( -prime) coatoms of C. Indeed, C(x) is the least member of C that is not contained in S − x. ✷
An easy verification shows that the -prime elements of C are exactly the point  closures C(m) of elements m ∈ S \ C that are κ-irreducible over C, i.e., m =  X with X ⊆ S implies m = b ∧ c for some b ∈ X and c ∈ C. κ S COROLLARY 3.1. The assignment x →
C(x) yields an isomorphism between the poset (S \ C,
C ) and the poset of all -irreducible elements of C. Under that isomorphism, those elements which are κ- -irreducible over C correspond to the
-prime elements of C. In particular, taking for C
the singleton {⊤} (where ⊤ is the top element of S), we see that the poset of all -irreducibles of the lattice Sκ is isomorphic to the antichain (S − ⊤, =); indeed, Sκ is atomistic with atoms {x, ⊤} (x ∈ S − ⊤). INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS 143 Next, we recall some definitions involving the covering relation (see [4] for the dual notions). A lattice is lower semimodular if a ≺ a ∨ b implies a ∧ b ≺ b. A complete lattice is lower locally distributive (respectively, lower locally Boolean)  if for each element b, the interval [ {a : a ≺ b}, b] is distributive (respectively, Boolean). In the Boolean case, such an interval  is isomorphic to the power set of {a : a ≺ b}. If for any a ≺ b, the element a is -prime in the interval (b] of all c
b, we speak of a local coframe, because the dual condition is certainly fulfilled in every frame. Weak atomicity of a lattice (or poset) means that each interval [a, b] with a < b contains a covering pair. Similarly, coatomicity of [a, b] means that each subinterval [c, b] with a
c < b contains a coatom, that is, an element covered by b. A lattice is strongly coatomic if each of its intervals is coatomic. As a consequence of the corresponding results for extremally detachable closure systems (cf. [10]), we note: THEOREM 3.1. Each interval C of κ-subsemilattices is a lower semimodular
local coframe in which every interval is -irreducibly generated. In particular, C is lower locally Boolean and weakly atomic. These claims are easily checked with the help of Lemma 3.2. For example, in order to see that C is lower locally Boolean, consider a fixed member B ∈ C, put A = B \(Mκ B)\C) and remark that any set Y with A ⊆ Y ⊆ B is again a member  of C (it suffices to verify κ-meet-closedness of Y : given X ⊆κ Y and a = S X, we must have a ∈ Y because a ∈ B \ Y ⊆ Mκ B would lead to the contradiction a ∈ X ⊆ Y ). Hence, the interval [A, B] is isomorphic to the power set of B \ A. But A is the intersection of all members of C that are covered by B. It is not hard to characterize those intervals C which are even Boolean. THEOREM 3.2. The following statements are equivalent for C = [C, S]:  (a) Each element of S \ C is κ- -irreducible in S. (b) C = {A : C ⊆ A ⊆ S}. (c) C is isomorphic to the power set of S \ C. (d) C is a (complete atomic)
Boolean lattice. (e) C is a frame whose -irreducible elements are incomparable. (f) The least element C is a meet of coatoms in C.  Proof. (a) ⇒ (b) For any subset Y of S, the closure is ŴY = Y ∪ C, by κ- irreducibility of all elements in Y \ C ⊆ S \ C. Thus C = {ŴY : Y ⊆ S} = {Y ∪ C : Y ⊆ S} = {A : C ⊆ A ⊆ S}. The implications (b) ⇒ (c) ⇒ (d) ⇒ (e) are clear. For (a) ⇔ (f) use Lemma 3.3.

(e) ⇒ (a) In a frame, -irreducible elements are -prime. Hence, a ∈ S \ C,
 X ⊆κ S and a = S X imply C(a) ⊆ ŴX = {C(x) : x ∈ X \ C} and then 144 MARCEL ERNÉ C(a) ⊆ C(b) for some b ∈ X\ C. Since C(a) and C(b) are comparable, a must coincide with b. Thus, a is κ- -irreducible in S. Finally, condition (d) is equivalent to (f) because C is lower locally Boolean. ✷ The (dualized) decomposition theorems due to Crawley and Dilworth [4] do not always apply to the intervals C, by lack of lower continuity. Recall that a complete lattice is lower continuous or join-continuous if for any  element  a and any chain (or down-directed subset) D, the distributive law a ∨ D = {a ∨ d : d ∈ D} is fulfilled. But this law is violated, for example, in the lattice C = Sω of all meetsubsemilattices of the distributive lattice L obtained by adding a top element ⊤ to the product {0, 1} × ω. For the meet-closed chains A = ({1} × ω) ∪ {⊤} and Di = {(0, n) : i
n ∈ ω}∪{⊤}, one computes A∨Di = L (since (0, n) = (1, n)∧(0, i) for n < i), but A ∨ {Di : i ∈ ω} = A. For a characterization of lower continuity of the subsemilattice systems Sω , see the end of this note. It should be mentioned that lower continuous, strongly coatomic local coframes have all the properties listed in Theorem 3.1 (see Diercks [5] for the dual statement, or Crawley and Dilworth [4] for the more restricted setting of algebraic lattices). Unfortunately, none of these implications remains true when one of the three hypotheses is dropped. Nevertheless, the unique decomposition theorem for lower continuous, strongly coatomic and lower locally distributive lattices has a strong counterpart for intervals of κ-meet-closed subsets. THEOREM 3.3. The following statements are equivalent for C = [C, S]: (a) (b) (c) (d) (e) C is coatomic.  S is κ- -irreducibly generated over C, i.e. S = Ŵ(Mκ S). S has a least join-decomposition in C, namely {C(m) : m ∈ Mκ S \ C}. S has a unique irredundant join-decomposition in C. S has an irredundant join-decomposition in C. Proof. Explicitly, condition (b) means that each element  of S is the meet of one element of C and fewer than κ elements of S that are κ- -irreducible. (a) ⇒ (b) Assume Ŵ(Mκ S) = S. Then Ŵ(Mκ S) is contained in a coatom A of C, and Lemma 3.3 applies, saying that A is of the form S − m for some element m ∈ Mκ S \ A. But then m ∈ Ŵ(Mκ S) ⊆ A, a contradiction.
(b) ⇒ (c) By Lemma 2.2, the sets C(m) with m ∈ Mκ S \ C are -irreducible in C (in fact, they are even superprime by Lemma 3.3), and  {C(m) : m ∈ Mκ S \ C}. S = Ŵ(Mκ S) = Ŵ(Mκ S \ C) = C Let D be an arbitrary join-decomposition of S in C. Again
by Lemma 2.2, we have D = {C(y) : y ∈ Y } for some Y ⊆ S \ C, and S = C D = ŴY . Any  m ∈ Mκ S \ C satisfies m∈ S = ŴY , hence m = c ∧ S X for an element c ∈ C / C), and and a set X ⊆κ Y . By κ- -irreducibility of m, this entails m ∈ Y (as m ∈ INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS 145 therefore C(m) ∈ D. In all, this shows that {C(m) : m ∈ Mκ S \ C} is contained in every join-decomposition of S and is, therefore, the least one. The implications (c) ⇒ (d) ⇒ (e) are trivial.
(e) ⇒ (a) S has an irredundant decomposition, S = C {C(y) : y ∈ Y } = ŴY , and no X ⊂ Y satisfies S = ŴX. If C ⊆ A ⊂ S for some A ∈ C then choose m ∈ Y \ A (such an element m exists because otherwise Y ⊆ A and S = ŴY ⊆  A). We show that S − m is a coatom in C containing A. The assumption m = S X for a set X ⊆κ S − m would  lead to m ∈ Ŵ(Y − m). Indeed, each x ∈ X has a meet representation x = cx ∧ S Ax with  cx ∈ C and Ax ⊆κ Y − m (note that m < x), whence x ∈ Ŵ(Y − m) and m = S X ∈ Ŵ(Y − m), by κ-meet-closedness of that set. But m ∈ Ŵ(Y − m) would already entail S = Ŵ(Y − m), which was excluded. ✷ Thus, m ∈ Mκ S \ C, and Lemma 3.3 applies again. Although Theorem 3.3 cannot be derived from decomposition theorems for lower continuous lattices, there is a local decomposition theorem, due to Diercks [5], that applies to our intervals C (see also Semyonova [14] for a similar result):
The top element of a -decomposition lattice L has a least join-decomposition if and only if L has the following three properties: (1) L is coatomic,  (2) Each coatom of L is -prime, (3) L is lower semimodular. As the required hypotheses (2) and (3) are fulfilled by the intervals C (see Theorem 3.1), one deduces at once the equivalence (a) ⇔ (c) in Theorem 3.3. However, the direct proof given before is instructive and shows more in the present context. COROLLARY 3.2. The following are equivalent for any interval C of κ-subsemilattices: (a) (b) (c) (d) (e) C is strongly coatomic.  Each member of C is κ- -irreducibly generated over C = C. Each member of C has a least join-decomposition. Each member of C has a unique irredundant join-decomposition. Each member of C has an irredundant join-decomposition. Gorbunov studied in [12] (see also [14, 15]) so-called canonical decompositions in complete lattices; these are
irredundant
join-decompositions D such that for each d ∈ D and any set X with D = X, there is an x ∈ X with d
x. Since
least decompositions in -irreducibly generated lattices are canonical, the above equivalent statements may be supplemented by (c′ ) Each member of C has a canonical join-decomposition. By Lemma 3.2, a further equivalent property is the following: (a′ ) For A, B ∈ C with A ⊂ B, there is an x ∈ B \ A with B − x ∈ C. 146 MARCEL ERNÉ Closure systems C with that property are called shellable and investigated more thoroughly in [10] (see also [6] for the finitary and [13] for the algebraic
case). Note that the superprimes of any shellable closure system are its maximal -primes. COROLLARY 3.3. If S is a lattice satisfying the ascending chain condition (requiring that each ascending sequence of elements becomes stationary) then each interval of Sκ is shellable and has the equivalent properties in Corollary 3.2. In fact, the ascending chain condition for S guarantees shellability of C = [C, S]: if A, B are members of Cwith A ⊂ B and x is maximal in B \ A then B − x belongs to C, because x is -irreducible in B. 4. Distributivity, Modularity and Upper Semimodularity Next, let us establish necessary and sufficient conditions for distributivity, respectively, modularity of the intervals C. Below, S7∗ denotes any 7-element lattice obtained by deleting one coatom from an 8-element Boolean lattice. This is the smallest lower but not upper semimodular lattice. We say a lattice contains a sublattice S7∗ faithfully if the three “vertical” cover relations hold not only in S7∗ but also in the entire lattice. The exclusion of that situation is characteristic for certain weak distributivity properties (cf. [5]). Recall that a lattice L is said to be meet-semidistributive if for all a, b1 , b2 , c ∈ L, the equation a ∧b1 = a ∧b2 = c entails a ∧ (b1 ∨ b2 ) = c. THEOREM 4.1. The following statements are equivalent: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) O = {S \ B : B ∈ C} is a topology (on S \ C). C is closed under binary unions. C is a coframe (i.e. a dual frame). C is distributive. C is modular. C is meet-semidistributive. All atoms of any subinterval C ′ of C are ∨-prime in C ′ . C does not contain any sublattice S7∗ (faithfully). a ∧ b ∈ C(a) ∪ C(b) for all a, b ∈ S. For distinct a, b, s ∈ S \ C with s = a ∧ b, there is a c ∈ C with s = a ∧ c or s = b ∧ c. 147 INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS Each of these conditions implies (k) C is upper semimodular and lower continuous. If C is shellable (strongly coatomic), the converse implication is also true. Proof. The implications (a) ⇔ (b) ⇒ (c) ⇒ (d) ⇒ (e) ⇒ (h) and (d) ⇒ (f) ⇒ (g) ⇒ (h) and the equivalence (i) ⇔ (j) are straightforward. (h) ⇒ (i) Assume a, b, s are distinct elements of S \ C with s = a ∧ b but not s ∈ C(a) ∪ C(b). We show that under these hypotheses, C contains the following sublattice S7∗ faithfully: C(a) ∨ C(b) C(s) ∪ C(a) B ∪ C(a) C(s) ∪ C(b) C(s) B ∪ C(b) B = C(s) − s First, we claim that C(s) ∪ C(a) and C(s) ∪ C(b) are elements of C. For this, note that each set U ⊆κ C(s) ∪ C(a) is of the form U = V ∪ {s ∧ w : w ∈ W } ∪ {a ∧ x : x ∈ X} with Z = V ∪ W ∪ X ⊆κ C.   For c = S Z ∈ C, it follows that S U ∈ {c, s ∧ c, a ∧ c} ⊆ C(s) ∪ C(a) (because s = a ∧ b = a ∧ s). Hence, C(s) ∪ C(a) is κ-meet-closed, and by analogy, so is C(s) ∪ C(b). We already know that B = C(s) − s is in C. But s is not the meet of less than κ elements in B ∪ C(a) or in B ∪ C(b) (otherwise, it would belong to C(a) or to C(b), as x < s for x ∈ B \ C), and consequently both B ∪ C(a) = C(s) ∪ C(a) − s and B ∪ C(b) = C(s) ∪ C(b) − s are in C, too. Furthermore, (C(s) ∪ C(a)) ∩ (C(s) ∪ C(b)) = C(s) ∪ (C(a) ∩ C(b)) = C(s), since c = a ∧ x = b ∧ y ∈ C(a) ∩ C(b) with x, y ∈ C entails c = a ∧ b ∧ x ∧ y = s ∧ x ∧ y ∈ C(s). Now, we conclude that (B ∪ C(a)) ∩ (C(s) ∪ C(b)) = B = (C(s) ∪ C(a)) ∩ (B ∪ C(b)) because s is not in B ∪ C(a) ∪ C(b). The other joins and meets in S7∗ are clear. Finally, the proper inclusion chains 148 MARCEL ERNÉ B ⊂ B ∪ C(a) ⊂ C(s) ∪ C(a) ⊂ C(a) ∨ C(b), B ⊂ B ∪ C(b) ⊂ C(s) ∪ C(b) ⊂ C(a) ∨ C(b), B ⊂ C(s) ⊂ C(s) ∪ C(a) and C(s) ⊂ C(s) ∪ C(b) are witnessed by the memberships s ∈ C(s) \ (C(a) ∪ C(b)), a ∈ C(a) \ (C(b) ∪ C(s)), b ∈ C(b) \ (C(a) ∪ C(s)). the form Z = X ∪ Y with X ⊆κ A (i) ⇒ (b) Let A, B ∈ C. AnyZ ⊆κ A ∪ B is of and Y ⊆κ B, whence a = S X ∈ A, b = S Y ∈ B, and S X = a ∧ b ∈ C(a) ∪ C(b) ⊆ A ∪ B. This shows that A ∪ B is κ-meet-closed and therefore a member of C. For the implication (e) ⇒ (k), it suffices to remark that  by Theorem 3.1, C is weakly atomic and the coatoms of any subinterval are -prime in that interval. Every modular lattice with these two properties is lower continuous. Conversely, a strongly coatomic, lower continuous, upper and lower semimodular lattice is already modular (see [5, II.3.1 and II.4.4] for the dual statements). ✷ Using the excluded pentagon characterization, one easily proves the equivalence of modularity and distributivity for the much larger class of weakly atomic local coframes. An obvious question is now: Are the properties in Theorem 4.1 equivalent to (upper) semimodularity? For finite C, the answer is certainly in the affirmative, because then upper and lower semimodularity together already entail modularity. However, for infinite C, that conclusion fails in general, as will be shown by the following counterexample, which is also useful in other contexts (see [4, 5, 14]): EXAMPLE 4.1. Consider a complete lattice S with the diagram below: INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS 149
The “middle” chain C of all non-minimal -reducible elements is clearly a sublattice of S, and the interval C = [C, S] in Sω has the following diagram: From the picture, one reads off the following properties of the interval C: it is • • • • algebraic and strongly coatomic, satisfying the ascending chain condition, upper and lower semimodular, upper but not lower continuous, and each element has a finite least join-decomposition. But C contains many sublattices S7∗ faithfully and is therefore not modular. Furthermore, C is not atomic, and the only atom is not ∨-prime. In view of the preceding example, we are interested in a general characterization of those intervals C = [C, S] which are semimodular (of course, such a characterization should be given by means of their least and greatest elements, C and S, respectively). To that aim, we introduce an appropriate density property and say a subset C of a meet-semilattice S is weakly meet-dense if for all a, b ∈ S with a 
b, there is a c ∈ C with a ∧ b = b ∧ c or a ∧ b
a ∧ c < a. Clearly, every meet-dense subset has that property (satisfying the stronger condition that for a 
b, there is a c ∈ C with a 
c but b
c, hence a ∧ b
a ∧ c < a). THEOREM 4.2. The following statements are equivalent: (a) (b) (c) (d) (e) (f) C is upper semimodular. A, B ∈ C and A + x ∈ C imply A ∨ B + x ∈ C. A, B ∈ C and x ∈ S \ A imply A(x) ∨ B − x ⊆ (A(x) − x) ∨ B. C(a) ∨ C(b) − a ⊆ (C(a) − a) ∨ C(b) for all a, b ∈ S. C is weakly meet-dense in S. For distinct a, b, s ∈ S \ C with s = a ∧ b, there is a c ∈ C with s = a ∧ c or s = b ∧ c or (s
a ∧ c < a and s
b ∧ c < b). Proof. The equivalence of the first three statements is easily verified (using Lemma 3.2) and holds in arbitrary extremally detachable systems. 150 MARCEL ERNÉ (c) ⇒ (d) Put A = C, B = C(b), x = a. If x ∈ A then C(a)∨C(b)−a ⊆ C(b), while for x ∈ / A, we get C(a) ∨ C(b) − a = A(x) ∨ B − x ⊆ (A(x) − x) ∨ B = (C(a) − a) ∨ C(b). (d) ⇒ (e) Suppose a, b ∈ S and a 
b. Then we have a∧b ∈ C(a)∨C(b)−a ⊆ (C(a) − a) ∨ C(b). Thus, we find a c ∈ C with a ∧ b = b ∧ c ∈ C(b) or a ∧ b = a ∧ c ∧ b and a ∧ c ∈ C(a) − a, hence a ∧ b
a ∧ c < a. (e) ⇒ (f) If a ∧ c = a ∧ b = s = b ∧ c for all c ∈ C then weak meet-density yields elements c1 , c2 ∈ C such that s
a ∧ c1 < a and s
b ∧ c2 < b. For c = c1 ∧ c2 ∈ C, it follows that s
a ∧ c < a and s
b ∧ c < b. (f) ⇒ (e) Suppose a 
b. If b
a then the top element ⊤ belongs to C and satisfies a ∧ b = b = b ∧ ⊤; if not, a and b are incomparable, and we may assume that none of the elements a, b, s = a ∧ b belongs to C. Now, we find a c ∈ C with s = a ∧ c or s = b ∧ c or (s
a ∧ c < a and s
b ∧ c < b). But in the first case, the inequality a
c would imply a = a ∧ b
b. Thus, s = a ∧ c < a in that case. (e) ⇒ (c) x ∈ / A ensures the inclusion A ∨ B ⊆ (A(x) − x) ∨ B. Suppose now that y ∈ A(x) ∨ B − x but y ∈ / A ∨ B. Then there exist elements a ∈ A, b ∈ B and c ∈ C such that y = a ∧ b ∧ c ∧ x < x. For a ′ = x and b′ = a ∧ b ∧ c we obtain a ′ 
b′ . Hence, by weak meet-density, there is a c′ ∈ C with y = a ′ ∧b′ = b′ ∧c′ = a∧b∧c∧c′ ∈ A∨B (which was excluded), or y = a ′ ∧b′ ∧c′ = (a∧x ∧c′ )∧(b∧c) and a ∧ x ∧ c′
a ′ ∧ c′ < a ′ = x. But then a ∧ x ∧ c′ ∈ A(x) − x, b ∧ c ∈ B, and so y ∈ (A(x) − x) ∨ B. ✷ Let us note two immediate consequences: COROLLARY 4.1. If C is meet-dense in S then the interval C = [C, S] is upper semimodular. COROLLARY 4.2. If all elements of C are ∧-prime in S then upper semimodularity of C already implies distributivity. Indeed, the alternative s = a ∧ b
a ∧ c < a and s
b ∧ c < b in (f) means a 
c, b 
c, but a ∧ b
c. Compare condition (j) in Theorem 4.1 with condition (f) in Theorem 4.2! It is not evident that these two conditions are equivalent under certain finiteness hypotheses. If C − ⊤ is a down-set, they cannot be fulfilled unless S is a chain (which excludes the existence of pairwise distinct elements a, b, a ∧ b). In particular, Sκ is distributive ⇔ Sκ is upper semimodular ⇔ S is a chain. A further immediate consequence of Theorem 4.1 is: COROLLARY 4.3. If every element of S is either ∧-irreducible, or a member of C, or covered by some c ∈ C, then C = [C, S] is distributive. INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS 151 Weak semimodularity, requiring that a ∧ b ≺ a, b implies a, b ≺ a ∨ b, may be described in our specific setting as follows: LEMMA 4.1. C is weakly semimodular if and only if for all C ′ ∈ C and all incomparable a, b in S\C ′ with a∧b ∈ / C ′ , there is a c ∈ C ′ such that a∧c ∈ / C ′ +a ′ or b ∧ c ∈ / C + b. The proof is an easy exercise, using Lemma 3.2 once again. Note that under suitable finiteness conditions (or strong atomicity together with upper continuity), weak semimodularity is equivalent to upper semimodularity (see [4]). By similar methods as in the proof
of Theorem 4.1, one proves an infinitary variant, involving the notions of κ- -prime elements and κ-frames. THEOREM 4.3. The following conditions on C are equivalent: (a) (b) (c) (cκ ) (d) (dκ ) (e) X ⊆κ O = {S \ B : B ∈ C} implies X ∈ O. C is closed under unions of nonempty subsystems Y ⊆κ C. C is an A-lattice. C is a
κ-frame.
Each
-irreducible element of C is
-prime. Each -irreducible elementof C is κ- -prime. Each element of S \ C is κ- -irreducible over C. Proof. The implications (a) ⇔ (b) ⇒ (d) ⇒ (c) ⇒ (cκ ) ⇒ (dκ ) and (c) ⇒ (d) ⇒ (dκ ) are clear. (dκ ) ⇒ (e) Suppose
that a ∈ S \ C is the meet of a set
X ⊆κ S. Then we obtain
C(a) ⊆ ŴX = C {C(x) : x ∈ X}, and as C(a) is -irreducible, hence κ- -prime, it must be contained in C(b) for some b ∈ X, i.e. a = b ∧ c for a suitable c ∈ C.  (e) ⇒ (b) Let ∅ = Y ⊆κ C and B = Y. As Y is nonempty,  B contains C. ∈ Y, we have X ∩ Y ⊆κ Y and therefore x = For X ⊆κ B and Y  Y S (X ∩ Y ) ∈ Y .  It follows that a = S X coincides with the meet S {xY : Y ∈ Y}. By hypothesis (e), a ∈ S \ Centails a = xY ∧ c for some Y ∈ Y and c ∈ C, whence a ∈ Y . We see that B = Y is κ-meet-closed. Thus B ∈ C. ✷ For κ = ∞, the last theorem amounts to: COROLLARY 4.4. Let C be a meet-closed subset of a complete lattice S. Then the following conditions on the interval C = [C, S] in S∞ are equivalent: (a) (b) (c) (d) (e) O = {S \ B : B ∈ C} is an Alexandroff topology. C is isomorphic to the Alexandroff completion of the poset (S \ C,
C ). C is an
A-lattice.
Each -irreducible elementof C is -prime. a ∈ S \ C, X ⊆ S and a = S X imply a = x ∧ c for some x ∈ X, c ∈ C. 152 MARCEL ERNÉ Some of the previous results generalize or correct statements for a specific finite case treated in [16]. The Alexandroff completion of a poset consists of all downsets A = ↓A = {b : b
a for some a ∈ A} (see, for example, [2] or [9]). We conclude our considerations with a remark concerning the case κ = ω. For a meet-semilattice S, the closure system Sω of all meet-subsemilattices is obviously algebraic, but it need not be coalgebraic (that is, the dual of Sω need not be an algebraic lattice) and not even lower continuous (join-continuous). A nice characterization of semilattices with coalgebraic Sω , by means of two excluded subsemilattices, was given by Adaricheva [1]. Since the proof in [1] is rather long and complicated, with a minor gap at the end, and since the english translation is erroneous or misleading at several places, we provide here a short alternative proof. By a hook in a meet-semilattice, we mean a subsemilattice C ∪ {a, b} where C is isomorphic to ω or its dual, and a = b ∧ c < b, c for all c ∈ C. THEOREM 4.4. For a meet-semilattice S, the following are equivalent: (a) Sω is coalgebraic. (b) Sω is lower continuous. (c) S does not contain any hook. Proof. (a) ⇒ (b) is well-known and easy (see [4] for the dual). (b) ⇒ (c) If there would exist a hook C ∪ {a, b} in S, we would obtain an infinite decreasing sequence of subchains Cn ⊆ C satisfying {Cn : n ∈ ω} = ∅ but {a, b} ⊆ Cn ∨ {b} for any n ∈ ω, whence {Cn : n ∈ ω} ∨ {b} = {b} = {Cn ∨ {b} : n ∈ ω}. (c) ⇒ (a) For X ∈ Sω and a ∈ S \ X, we have to find a cocompact Y ∈ Sω with X ⊆ Y and a ∈ / Y (then X is the intersection of cocompact members of Sω ). Put [a) = {b ∈ S : a
b} and ]a) = {b ∈ S : a < b}. By Zorn’s Lemma, there is a maximal filter F in ]a) containing ]a) ∩ X. We claim that Y = F ∪ (S \ [a)) does the job. Clearly, Y is a subsemilattice containing X but not a. Its complement Z = S \ Y = [a) \ F cannot contain any infinite chain. Indeed, let C be a maximal nonempty chain in Z − a. There is a b ∈ C and an e ∈ F with a = b ∧ e (otherwise, C ∪ F would generate a filter in ]a) that properly contains F ). Any chain D in F ∩ (e] must be finite, since a = b ∧ d for any d ∈ D and there is no hook in S. Hence, F has a least element f , which covers a and satisfies a = c ∧ f for any c ∈ C. Consequently, C must be finite. It follows that all nonempty subsets of Z (not only the finite ones) have infima in Z. Now, it is easy to see that Y is cocompact: if  D is a chain in Sω such that D ⊆ Y for all D ∈ D then the elements zD := (D \ Y ) form a chain in Z; hence, there must be a greatest among them, say zD0 , and this element belongs to D \ Y . Indeed, for D1 ∈ D, there is a D ∈ D with D ⊆ D0 ∩ D1 , whence ✷ zD 0 = zD ∈ D 1 . INTERVALS IN LATTICES OF κ-MEET-CLOSED SUBSETS 153 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Adaricheva, K. V.: Semidistributive and coalgebraic lattices of subsemilattices, Algebra and Logic 27 (1988), 385–395. Alexandroff, P.: Diskrete Räume, Mat. Sb. 2 (1937), 501–518. Birkhoff, G.: Lattice Theory, 3rd edn, Amer. Math. Soc. Publ. XX, Providence, RI, 1967. Crawley, P. and Dilworth, R. P.: Algebraic Theory of Lattices, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. Diercks, V.: Zerlegungstheorie in vollständigen Verbänden, Diplomarbeit (Thesis), University of Hannover, 1982. Edelman, P. H. and Jamison, R. E.: The theory of convex geometries, Geom. Dedicata 19 (1985), 247–270. Erné, M.: Ordnungs- und Verbandstheorie, Fernuniversität Hagen, 1985. Erné, M.: Bigeneration in complete lattices and principal separation in ordered sets, Order 8 (1991), 197–221. Erné, M.: Algebraic ordered sets and their generalizations, In: I. Rosenberg and G. Sabidussi (eds), Algebras and Orders, Proc. Montreal, 1992, Kluwer Acad. Publ., Amsterdam, 1993. Erné, M.: Convex geometries and anti-exchange properties, Preprint, University of Hannover, 2002. Erné, M., Šešelja, B. and Tepavčević, A.: Posets generated by irreducible elements, Order 20 (2003), 79–89. Gorbunov, V.: Canonical decompositions in complete lattices, Algebra and Logic 17 (1978), 323–332. Jamison-Waldner, R.: Copoints in antimatroids, In: Proc. 11th SE Conf. Comb., Graph Theory and Comput., Congressus Numerantium 29 (1980), 535–544. Semyonova, V. M.: Lattices with unique irreducible decompositions, Algebra and Logic 39 (2000), 54–60. Semyonova, V. M.: Decompositions in complete lattices, Algebra and Logic 40 (2001), 384– 390. Šešelja, B. and Tepavčević, A.: Collection of finite lattices generated by a poset, Order 17 (2000), 129–139.