Lattices of relative colour-families and antivarieties

Discussiones Mathematicae General Algebra and Applications 27 (2007 ) 123–139 LATTICES OF RELATIVE COLOUR-FAMILIES AND ANTIVARIETIES Aleksandr Kravchenko Sobolev Institute of Mathematics SB RAS Novosibirsk, Russia e-mail: [email protected] Abstract We consider general properties of lattices of relative colour-families and antivarieties. Several results generalise the corresponding assertions about colour-families of undirected loopless graphs, see [1]. Conditions are indicated under which relative colour-families form a lattice. We prove that such a lattice is distributive. In the class of lattices of antivarieties of relation structures of finite signature, we distinguish the most complicated (universal) objects. Meet decompositions in lattices of colour-families are considered. A criterion is found for existence of irredundant meet decompositions. A connection is found between meet decompositions and bases for anti-identities. Keywords: colour-family, antivariety, lattice of antivarieties, meet decomposition, basis for anti-identities. 2000 Mathematics Subject Classification: 08C15. 1. Preliminary facts Throughout the article, by a structure we mean a relation structure of a fixed signature σ = (rj )j∈J . A structure is said to be finite if its universe is a finite set. A homomorphism from a structure A into a structure B is a map ϕ : A → B such that (ϕ(a1 ), . . . , ϕ(an )) ∈ rjB for all j ∈ J and The work was partially supported by the INTAS (grant 03-51-4110) and the Russian Council for Support of Leading Scientific Schools (grant NSh-4413.2006.1). 124 A. Kravchenko a1 , . . . , an ∈ A with (a1 , . . . , an ) ∈ rjA. If there exists a homomorphism from A into B then we write A → B; otherwise, we write A 9 B. For every class K, let Kf denote the set of isomorphism types of finite structures in K. On Kf , define an equivalence relation ≡ as follows: A ≡ B if and only if A → B and B → A. The relation → induces a partial order ≤ on the quotient set Kf /≡ . Let Core(K) denote the resulting partially ordered set. In the sequel, it is convenient to consider an isomorphic partially ordered set whose universe is the set of cores of finite structures in K. Recall [2, Section 2] that a finite structure is a core if all its endomorphisms are automorphisms. A structure A is a core of a structure B if A is a minimal retract of B (with respect to set inclusion). Simple properties of cores can be found, for example, in [2, Proposition 2.1]. It is easy to see that, in every coset G/≡ , there exists a unique (up to isomorphism) core. We denote this core by Core(G). The map defined by the rule G/≡ 7→ Core(G) is an isomorphism. Let K be a class of structures. For every A ∈ K, let [K → A] = {B ∈ K : B → A}. If there is no ambiguity or K is the class of all structures of a given signature then we S write [→ A] instead of [K → A]. For every set A ⊆ K, let [K → A] = A∈A [K → A]. If A is a finite set of finite structures then [K → A] is called a K-colour-family. If |A| = 1 then the K-colour-family [K → A] is said to be principal. Let L0 (K) denote the partially ordered set of all K-colour-families with respect to set inclusion. (If L 0 (K) has no greatest element then by L0 (K) we mean the set of all K-colour-families with a new greatest element 1K ). We recall the definition of operations with structures from [3]. Let A and B be structures and let (Ai )i∈I be a family of structures. On the disjoint union of the universes of A and B, define a structure A + B of signature σ as follows: (a1 , . . . , an ) ∈ r A+B if and only if either (a1 , . . . , an ) ∈ r A or (a1 , . . . , an ) ∈ r B. The resulting structure is called the sum of A and B. We have (1) A + B → C ⇐⇒ A → C and B → C for every structure C. A structure A is said to be connected if it cannot be represented in the form A1 + A2 , where A 9 Ai , i = 1, 2. Lattices of relative colour-families and antivarieties 125 On a structure Q the Cartesian product of the universes of A i , i ∈QI, define i∈I Ai if and only if A of signature σ as follows: (a , . . . , a ) ∈ r 1 n i∈I i (a1 (i), . . . , an (i)) ∈ r Ai for every i ∈ I. The resulting structure is called the product of the family (Ai )i∈I . We have (2) C→ Q Ai ⇐⇒ C → Ai for all i ∈ I i∈I for every structure C. On the set AB of all functions from B into A, define a structure A B of B signature σ as follows: (f1 , . . . , fn ) ∈ r A if and only if (f1 (b1 ), . . . , fn (bn )) ∈ r A for all b1 , . . . , bn ∈ B with (b1 , . . . , bn ) ∈ r B. The resulting structure is called the exponent of A by B. We have C → AB ⇐⇒ B × C → A (3) for every structure C. 2. Lattices of colour-families In this section, we show that partially ordered sets of K-colour-families are usually lattices and study their lattice-theoretical properties. Lemma 1. If K is a class satisfying the condition (4) A × B ∈ K for all finite A, B ∈ K then L0 (K) is a distributive lattice with respect to the set-theoretical operations. P roof. Let K1 = [K → (Ai )i<n ], K2 = [K → (Bj )j<m ]. It is clear that K1 ∪ K2 is a K-colour-family, i.e., K1 ∨ K2 = K1 ∪ K2 . Let A ∈ K1 ∩ K2 . Then A ∈ K and there exist i < n and j < m such that A → Ai and A → Bj . Hence, A → Ai × Bj , cf. (2). By (4), we have Ai × Bj ∈ K. Conversely, if A → Ai × Bj , where A ∈ K, i < n, and j < m, then we   have A → Ai and A → Bj . Hence, K1 ∩ K2 = K → (Ai × Bj )i<n, . In view of (4), we have Ai × Bj ∈ K for all i < n j<m and j < m. 126 A. Kravchenko Remark 2. Another lattice of (principal) colour-families was considered in [4, 5]. In fact, the universe of that lattice is the set of cores, the meet operation coincides with the meet operation in L 0 (K), while the join operation corresponds to the sum of relation structures. That lattice is distributive too. Let L be a lattice and let a, b ∈ L. By a relative pseudocomplement of a with respect to b we mean an element a ∗ b such that a ∧ x 6 b ⇐⇒ x 6 a ∗ b for all x ∈ L. If a relative pseudocomplement exists for every pair of elements of L then L is said to be a relatively pseudocomplemented lattice. Lemma 3. If K is a class satisfying (4) and the condition (5) AB ∈ K for all finite A, B ∈ K with B 9 A then L0 (K) is a relatively pseudocomplemented lattice. P roof. Let K1 = [K → (Ahi )i<n ] and let Ki2 = [K → (Bj )j<m ]. We inAi i troduce the notation Ki = K → (BA j )j<m . By (5), we have Bj ∈ K provided Ai 9 Bj . If Ai h→ Bj theni Ai × C → Bj for every C ∈ K. By (3), Ai i = K is the greatest element of L0 (K). we obtain C → BA j , i.e., K → Bj i h Ai Therefore, K → Bj ∈ L0 (K) for all i < n and j < m. i h i By Lemma 1, we have [K → Ai ] ∩ Ki = K → (BA × A ) i j<m . It is j i i immediate from (3) that BA j × Ai → Bj . Hence, [K → Ai ] ∩ K ⊆ K2 . Let K3 = [K → (Ck )k<l ] be a K-colour-family such that [K → Ai ] ∩ K3 ⊆ K2 . By Lemma 1, we have [K → Ai ] ∩ K3 = [K → (Ai × Ck )k<l ]. Therefore, for every k < l, there exists a j < m such that A i × Ck → Bj . By definition, i i Ck → BA j . Thus, K3 ⊆ K . We have proven that Ki is a pseudocomplement of [K → Ai ] with respect to K2 . For every distributive lattice L and elements a, b, c ∈ L, if a ∗ c and b ∗ c exist then so does (a ∨ b) ∗ cTand (a ∨ b) ∗ c = (a ∗ c) ∧ (b ∗ c), cf. [6, Theorem 9.2.3]. Hence, K1 ∗ K2 = i<n Ki . Lattices of relative colour-families and antivarieties 3. 127 Lattices of antivarieties Recall [2] that a K-antivariety is a class defined in K by some (possibly, empty) set of anti-identities, i.e., sentences of the form ∀x1 . . . ∀xn (¬ R1 (x) ∨ · · · ∨ ¬ Rm (x)), where each Ri (x) is an atomic formula. By [2, Theorem 1.2], for every universal Horn class K, a subclass K ′ is a K-antivariety if and only if K′ = K ∩ H−1 SP∗u (K′ ), where H−1 , S, and P∗u are operators for taking homomorphic pre-images, substructures, and nontrivial ultraproducts. In particular, each K-colour-family is a K-antivariety. Let L(K) denote the partially ordered set of all K-antivarieties with respect to set inclusion. Lemma 4. For every universal Horn class K, the partially ordered set L(K) is a distributive lattice with respect to the set-theoretical operations. P roof. It is clear that K1 ∧ K2 = K1 ∩ K2 and K1 ∪ K2 ⊆ K1 ∨ K2 for all K1 , K2 ∈ L(K). Since K1 and K2 are elementary classes, the class K1 ∪ K2 is elementary too. In particular, P∗u (K1 ∪ K2 ) ⊆ K1 ∪ K2 . By [2, Theorem 1.2], we have K1 ∨ K2 = K ∩ H−1SP∗u (K1 ∪ K2 ). Hence, for every A ∈ K1 ∨ K2 , there exists a B ∈ K1 ∪ K2 such that A → B. Since K1 and K2 are closed under H−1 S in K, we obtain A ∈ K1 ∪ K2 and, consequently, K1 ∨ K 2 = K 1 ∪ K 2 . The proof of the following lemma is similar to that of [1, Lemma 3.4] Lemma 5. For every universal Horn class K and K-antivariety X, we have X = H−1 SP∗u (Core(X)) ∩ K. If σ is a finite signature then the partially ordered set Core(K) and the lattices L0 (K) and L(K) are related as follows. Lemma 6. For every universal Horn class K of finite signature, the lattice L(K) is isomorphic to the ideal lattice I(L 0 (K)) of the lattice L0 (K) and to the lattice Io (Core(K)) of order ideals of the partially ordered set Core(K). This lemma generalises [1, Theorem 3.6] to the case of arbitrary relation structures of finite signature. The proof follows the lines of the proof in [1]. 128 A. Kravchenko P roof. W Put ϕ(K′ ) = {A ∈ L0 (K) : A ⊆ K′ } for every K′ ∈ L(K) and put ψ(J) = {A ∈ L0 (K) : A ∈ J} for every ideal J of L0 (K). It is easy to see that ϕ is a map from L(K) into I(L0 (K)) and ψ is a map from I(L0 (K)) into L(K); moreover, both maps are monotone. We prove that ϕ = ψ −1 , which implies that ϕ and ψ are isomorphisms. It is clear that ϕψ(J) ⊇ J and ψϕ(K′ ) ⊆ K′ for all J ∈ I(L0 (K)) and ′ K ∈ L(K). ′ Let A ∈ K′ be a finite structure. Then K′ is a S [K → A] ⊆ K because K-antivariety. We have A ∈ [K → A] ⊆ {A ∈ L0 (K) : A ⊆ K′ }. Since the K-antivariety ψϕ(K′ ) is generated by its finite structures, we obtain ψϕ(K′ ) = K′ . W B. Then A = Let A ∈ ϕψ(J), i.e., A ∈ L0 (K) and A ⊆ B∈J W [K → (Ai )i<n ], where each finite structure Ai belongs to B∈J B.SBy [2, Theorem 1.2], for every i < n, there exist a family (B ij )j∈Ji ⊆ B∈J B Q and an ultrafilter Ui over Ji such that Ai → j∈Ji Bij . Since Ai is a finite structure of finite signature, from [1, Lemma 3.2] it follows that there   exists a j(i) ∈SJi such that Ai → Bj(i) . Hence, A ⊆ K → (B j(i) )i<n .  Since Bj(i) ∈ B∈J W B, we obtain  K → Bj(i) ∈ J. Since J is an ideal, we conclude that i<n K → Bj(i) ∈ J. Thus, A ∈ J and, consequently, ϕψ(J) = J. For proving the fact that L(K) and Io (Core(K)) are isomorphic put ϕ0 (K′ ) = {Core(A) : A ∈ K′ }, ψ0 (J) = H−1 SP∗u (J) ∩ K. By Lemma 5 and [1, Lemma 3.2], we have ϕ−1 0 = ψ0 . It is clear that ϕ0 and ψ0 are monotone. Therefore, the lattices L(K) and I o (Core(K)) are isomorphic. Corollary 7. For every universal Horn class K of finite signature, the lattice L(K) is relatively pseudocomplemented. For all K 1 , K2 ∈ L(K), the following equality holds: K1 ∗ K2 = H−1 SP∗u {A ∈ Kf : A × B ∈ K2 for all B ∈ (K1 )f } ∩ K. P roof. By [7, Corollary II.1.4], we have I ∗ J = {a ∈ L : a ∧ i ∈ J for all i ∈ I} for every distributive lattice L and ideals I and J of L. This equality, together with Lemma 6, yields the required assertion. Lattices of relative colour-families and antivarieties 4. 129 Complexity of lattices of antivarieties In this section, we introduce the notion of a universal (the most complicated) lattice among the lattices of antivarieties of relation structures of finite signature and give examples of universal lattices. Let K be a class of structures. By the category K we mean the category whose objects are structures in K and morphisms are homomorphisms. A one-to-one functor Φ from a category K 1 into a category K2 is called a full embedding if, for every morphism α : Φ(A) → Φ(B) in K 2 , there exists a morphism β : A → B in K1 such that Φ(β) = α. For more information about categories, the reader is referred to [8]. By G we denote the class (and the category) of undirected loopless graphs. Lemma 8. Let K be a class of structures and let there exist a full embedding Φ : G → K such that, for every finite graph G, the structure Φ(G) if finite. Then there exists an embedding ϕ : Core(G) → Core(K). P roof. Put ϕ(G) = Core(Φ(G)) for every G ∈ Core(G). It is clear that ϕ is a map from Core(G) into Core(K). We show that ϕ is an embedding. Let G ≤ H, where G, H ∈ Core(G), and let ψ be the corresponding homomorphism. Then the composition e Φ(ψ) r ϕ(G) = Core(Φ(G)) → Φ(G) −−−→ Φ(H) → Core(Φ(H)) = ϕ(H) is a homomorphism from ϕ(G) into ϕ(H). Hence, ϕ(G) ≤ ϕ(H). Let ϕ(G) ≤ ϕ(H) for some G, H ∈ Core(G) and let ψ be the corresponding homomorphism. Then the composition r ψ e Φ(G) → Core(Φ(G)) = ϕ(G) → ϕ(H) = Core(Φ(H)) → Φ(H) is a homomorphism from Φ(G) into Φ(H). Denote this homomorphism by α. Since Φ is a full embedding, we have α = Φ(β) for some homomorphism β : G → H. Hence, G ≤ H. It remains to show that ϕ is a one-to-one map. If ϕ(G) = ϕ(H) then ϕ(G) ≤ ϕ(H) and ϕ(H) ≤ ϕ(G). By the above, G ≤ H and H ≤ G. Hence, G = H. A category K satisfying the conditions of Lemma 8 is said to be finite-tofinite universal. As is known [9] (see also [10, Theorem 2.10]), the partially ordered set Core(G) is ω-universal, i.e., each countable partially ordered set 130 A. Kravchenko is embeddable into Core(G). By Lemma 8, for every finite-to-finite universal category K, the partially ordered set Core(K) is ω-universal. Recall that a lattice L is called a factor of a lattice K if L is a homomorphic image of a suitable sublattice of K. We say that L(K) is a universal lattice if, for every universal Horn class K ′ of relation structures of finite signature, the lattice L(K′ ) is a factor of the lattice L(K). Theorem 9. Let K be a universal Horn class of relation structures of finite signature. If K is a finite-to-finite universal category then L(K) is a universal lattice. P roof. By Lemma 8, for every universal Horn class K ′ of relation structures of finite signature, there exists an embedding ϕ : Core(K ′ ) → Core(K). By Lemma 6, we may consider the lattice of order ideals of the partially ordered set of cores instead of the lattice of antivarieties. For every I ∈ Io (Core(K′ )), put ψ(I) = {H ∈ Core(K) : H ≤ ϕ(G) for some G ∈ I}. It is easy to verify that, for every order ideal I of Core(K ′ ), the set ψ(I) is an order ideal of Core(K). We prove that ψ is one-to-one. Let ψ(I) = ψ(J). For every H ∈ I, we have ϕ(H) ∈ ψ(I) = ψ(J), i.e., there exists an element G ∈ J such that ϕ(H) ≤ ϕ(G). Since ϕ is an embedding, we have H ≤ G, i.e., H ∈ J. We have proven that I ⊆ J. The proof of the converse inclusion is similar. We prove that ψ is a join homomorphism. Consequently, the join semilattice of Io (Core(K′ )) is embeddable into the join semilattice of I o (Core(K)). The inclusion ψ(I)∨ψ(J) ⊆ ψ(I∨J) is obvious. Conversely, let H ∈ ψ(I∨J). Then there exists a G ∈ I ∨ J such that H ≤ ϕ(G). Since I ∨ J = I ∪ J, we obtain H ∈ ψ(I) ∪ ψ(J) = ψ(I) ∨ ψ(J). Let L be the sublattice of Io (K) generated by the set {ψ(I) : I ∈ Io (Core(K′ ))}. Then, for every X ∈ L, there exist a lattice term t(v0 , . . . , vn−1 ) and order ideals J0 , . . . , Jn−1 of Core(K′ ) such that X = t(ψ(J0 ), . . . , ψ(Jn−1 )). We prove that, for every lattice term t(v 0 , . . . , vn−1 ) and order ideals J0 , . . . , Jn−1 of Core(K′ ), the equality (6) t(ψ(J0 ), . . . , ψ(Jn−1 )) ∩ ϕ(Core(K′ )) = ϕ(t(J0 , . . . , Jn−1 )) holds, where ϕ(M ) = {ϕ(m) : m ∈ M } for every set M . Lattices of relative colour-families and antivarieties 131 We use induction on the length of the term. Let t(v 0 , . . . , vn−1 ) = vi . Then the right-hand side of (6) is ϕ(Ji ) and the left-hand side of (6) is ψ(Ji ) ∩ ϕ(Core(K′ )). It is clear that ϕ(Ji ) ⊆ ψ(Ji ) ∩ ϕ(Core(K′ )). Conversely, let H ∈ ψ(Ji ) ∩ ϕ(Core(K′ )). Then H = ϕ(G) for some G ∈ Core(K′ ). Let J be the least ideal of Core(K′ ) containing Ji ∪ {G}. We have ψ(J) = ψ(Ji ) ∪ {A ∈ Core(K′ ) : A ≤ ϕ(G)}. Since ϕ(G) = H ∈ ψ(Ji ), we obtain ψ(J) = ψ(Ji ). Since ψ is a one-to-one map, we have J = Ji , i.e., G ∈ Ji . Therefore, H ∈ ϕ(Ji ) ⊆ ψ(Ji ). Assume that t = t1 ∧ t2 or t = t1 ∨ t2 for some terms t1 and t2 . We introduce the notation Yi = ti (ψ(J0 ), . . . , ψ(Jn−1 )), Xi = ti (J0 , . . . , Jn−1 ), where i = 1, 2. By induction, Yi ∩ Core(K′ ) = ϕ(Xi ), i = 1, 2. If t = t1 ∧ t2 then t(ψ(J0 ), . . . , ψ(Jn−1 )) = Y1 ∩ Y2 , t(J0 , . . . , Jn−1 ) = X1 ∩X2 . By induction, Y1 ∩Y2 ∩ϕ(Core(K′ )) = ϕ(X1 )∩ϕ(X2 ) ⊇ ϕ(X1 ∩X2 ). For every A ∈ ϕ(X1 ) ∩ ϕ(X2 ), there exist Ai ∈ Xi , i = 1, 2, such that A = ϕ(A1 ) = ϕ(A2 ). Since ϕ is a one-to-one map, we obtain A 1 = A2 ∈ X1 ∩ X2 . It t = t1 ∨ t2 then t(ψ(J0 ), . . . , ψ(Jn−1 )) = Y1 ∪ Y2 , t(J0 , . . . , Jn−1 ) = X1 ∪ X2 . By induction, (Y1 ∪ Y2 ) ∩ ϕ(Core(K′ )) = (Y1 ∩ ϕ(Core(K′ ))) ∪ (Y2 ∩ ϕ(Core(K′ ))) = ϕ(X1 ) ∪ ϕ(X2 ) ⊆ ϕ(X1 ∨ X2 ). The converse inclusion is an easy consequence of the equality X1 ∨ X2 = X1 ∪ X2 . Since the operations of the lattice of order ideals are the set-theoretical operations, the union and the intersection, from (6) we obtain (7) ϕ−1 (t(ψ(J0 ), . . . , ψ(Jn−1 )) ∩ ϕ(Core(K′ ))) = t(J0 , . . . , Jn−1 ). Let X = t(ψ(J0 ), . . . , ψ(Jn−1 )) ∈ L. Put α(X) = ϕ−1 (X ∩ ϕ(Core(K′ ))). It is immediate from (7) that α is a map from L onto I o (Core(K′ )). By (6), α is a homomorphism. We present an example showing that the converse to Theorem 9 is not true. Namely, we indicate a quasivariety K of loopless digraphs such that K is not a finite-to-finite universal category but L(K) is a universal lattice. Example 10. Let σ consist of one binary relation symbol r. Denote by K the quasivariety of structures of the signature σ defined by the quasi-identities 132 A. Kravchenko
∀x∀y r(x, x) → x ≈ y ,
∀x∀y∀z r(x, y) & r(x, z) → y ≈ z ,
∀x∀y∀z r(y, x) & r(z, x) → y ≈ z . Let Cn , n > 2, denote the cycle of length n, i.e., the structure whose universe is Cn = {0, 1, . . . , n − 1} and (i, j) ∈ r Cn if and only if i + 1 ≡ j (mod n). It is easy to see that, for every n > 2, we have C n ∈ K. Let P denote the set of prime numbers. Denote by a p , p ∈ P, the K-colour-family [K → Cp ]. Let L be the sublattice of L0 (K) generated by the elements (ap )p∈P . We show that the distributive lattice L is freely generated by the set (ap )p∈P . Since |P| = ω, this means that the free distributive lattice F D (ω) of countable rank is embedded into L0 (K). We use [7, Theorem II.2.3]. It V suffices Wto verify that, for all finite nonempty subsets I, J ⊆ P, from j∈J aj it follows that I ∩ J 6= ∅. i∈I ai 6 V W Let I and J be finite and nonempty. Assume that i∈I ai 6 j∈J aj . By   W Q V Lemma 1, Qwe have i∈I ai = K → i∈I Q Ci and j∈J aj = [K → (Cj )j∈J ]. Let k = W is easy to see that i∈I Ci ≃ Ck (cf., for example, [11]). V i∈I i. It Since i∈I ai 6 j∈J aj , there exists a prime j ∈ J with Ck ∈ [K → Cj ]. We have Ck → Cj if and only if j divides k (cf., for example, [12]). Since j is prime and k is a product of distinct primes, we conclude that j ∈ I. Thus, I ∩ J 6= ∅. We show that the ideal lattice I(FD (ω)) of the free distributive lattice of countable rank is embeddable into L(K). Let L and K be distributive lattices and let ϕ : L → K be an embedding. Define a map ψ : I(L) → I(K) by the following rule: ψ(I) is the ideal of K generated by ϕ(I). Using the definition of an ideal generated by a set, we easily find that ψ is an embedding. In particular, I(FD (ω)) is embeddable into I(L0 (K)). The latter lattice is isomorphic to L(K) in view of Lemma 6. We show that the lattice L(G) of antivarieties of undirected loopless graphs is a homomorphic image of the lattice I(F D (ω)). Since L0 (G) is a countable distributive lattice, there exists a homomorphism from F D (ω) onto L0 (G). As above, this homomorphism induces a homomorphism between the corresponding ideal lattices. It remains to use Lemma 6. Lattices of relative colour-families and antivarieties 133 Therefore, L(G) is a factor of L(K). We conclude that L(K) is a universal lattice. The class of rigid objects in the category K consists of trivial structures and finite directed chains only (cf. [8, Exercise IV.1.6]). Therefore, the category K is not universal and, consequently, is not finite-to-finite universal. 5. Irredundant meet decompositions in lattices of colour-families Recall that G denotes the universal Horn class and the category of undirected loopless graphs. The study of the lattice L 0 (G) was initiated in [1]. It was proven that this lattice possesses neither completely join irreducible nor completely meet irreducible nonzero elements. A simple description for join irreducible colour-families was found. The question on meet irreducible elements turned to be closely connected with a well-known problem in the graph theory, Hedetniemi’s conjecture [13]. Here, we consider meet decompositions of K-colour-families with the help of Lemma 3, which says that the lattice of K-colour-families is relatively pseudocomplemented. A similar approach was first used in [4]. We present necessary definitions. By a meet decomposition of an element x ∈ L, where L is an arbitrary lattice, we mean a representation (8) x= V mi , i∈I where (mi )i∈I is a family of meet irreducible elements, i.e., for each i ∈ I, we have mi 6= 1 and from mi = a ∧ b it follows thatVeither mi = a or mi = b. A meet decomposition (8) is irredundant if x < i∈J mi for every proper subset J ( I. For distributive relatively pseudocomplemented lattices, the following criterion for meet irreducibility of elements is known [14]. Proposition 11. Let L be a distributive relatively pseudocomplemented lattice and let m ∈ L. The element m is meet irreducible if and only if x ∗ m = m for every x ∈ L with x m. Throughout this section, we assume that L is an arbitrary distributive relatively pseudocomplemented lattice. Let ∨, ∧, and ∗ denote the operations of L. For every x ∈ L, let Reg(x) = {y∗x : y ∈ L}, i.e., let Reg(x) denote the 134 A. Kravchenko set of regular elements of the principal filter [x) of L. For all u, v ∈ Reg(x), put u + v = ((u ∨ v) ∗ x) ∗ x, u · v = u ∧ v, 0 = x, 1 = 1L , u′ = u ∗ x. The set Reg(x) with the operations +, ·, and ′ and constants 1 and 0 is a Boolean algebra; moreover, the map r from L to Reg(x) defined by the rule r(y) = (y ∗ x) ∗ x is a homomorphism between Heyting algebras [6, Theorem 8.4.3]. We mention the following relationship between meet irreducible elements of L and dual atoms of Reg(x) [4, Theorem 6]. Proposition 12. Let x, y ∈ L and let x < y. The element y is a dual atom of the Boolean algebra Reg(x) if and only if y ∗ x > x and y is meet irreducible in L. Recall [15] that a Boolean algebra A is atomic if, for every nonzero element a ∈ A, there exists an atom b such that b 6 a. An element a is said to be atomless if a 6= 0 and there is no atom b such that b 6 a. Theorem 13. Let L be a distributive pseudocomplemented lattice and let a ∈ L. The element a admits an irredundant meet decomposition in L if and only if the Boolean algebra Reg(a) is atomic. V P roof. Let a = i∈I mi be an irredundant meet decomposition. We prove that mi ∗ a > a for every i ∈ I. In view of PropositionV 12, this means that each mi , V i ∈ I, is
a dual Since a = i∈I mi , we have V V atom of Reg(a). mi ∗ a = mi ∗ i∈I mi = mj mi mj = j6=i mj (cf. [16, IV.7.2 (8)]). Since V the meet decomposition is irredundant, we have m i ∗ a = j6=i mj > a. Assume that there exists an atomless element b ∈ Reg(a). Since the complement of a dual atom is an atom, we conclude that m i ∗ a b for all i ∈ I. Hence, b ∧ (mi ∗ a) = a for all i ∈ I. By the definition of a relative pseudocomplement, we have b 6 (mi ∗ a) ∗ a = mi for all i ∈ I. Therefore, V b 6 i∈I mi = a. This proves that Reg(a) possesses no atomless element, i.e., Reg(a) is an atomic Boolean algebra. Conversely, assume that Reg(a) is an atomic Boolean algebra. Let (ai )i∈I be the set of atoms. If | Reg(a)| = 2 then the element a is meet irreducible in view of Proposition 11. In the sequel, we assume that | Reg(a)| > 2. We denote mi = ai ∗ a, i ∈ I. For every i ∈ I, the element m i is a dual atom of Reg(a); moreover, each dual atom is of the form m i , i ∈ I. Lattices of relative colour-families and antivarieties 135 By Proposition 12, we have mi ∗ a > V a and mi is meet irreducible for every i ∈ I. It remains to show that a = i∈I mi (in the lattice L). It is clear that a is a lower bound for (mi )i∈I . If a is not the greatest lower bound then there exists a lower bound b0 ∈ L for (mi )i∈I such that b0 a. Consider the element (b0 ∗ a) ∗ a ∈ Reg(a). Since b0 6 mi , we conclude that (b0 ∗ a) ∗ a 6 (mi ∗ a) ∗ a = mi , i ∈ I. Hence, b = (b0 ∗ a) ∗ a ∈ Reg(a) is a lower bound for (mi )i∈I . Since b0 a and b0 6 b, we find that b 6= a, i.e., b > a. Since Reg(a) is an atomic Boolean algebra, there exists an atom aj such that aj 6 b. Passing to the complements, we find that b ′ 6 mj . Since b 6 mi for all i ∈ I, we obtain b + b′ = 1 6 mj + mj = mj < 1, a contradiction. Similar questions for undirected loopless graphs were considered in [17], where the notion of the level of nonmultiplicativity of a graph was introduced. In our terminology, the level of nonmultiplicativity of a graph G is the number of dual atoms of the Boolean algebra Reg([G → G]). In [17], the following conjecture is stated: The level of nonmultiplicativity of each finite graph is finite. In connection with Theorem 13, we formulate the following Problem 14. Let K be a universal Horn class of relation structures of finite signature. Is this true that, for every K-colour-family A, the following conditions are equivalent: V (1) there exists an irredundant meet decomposition A = i∈I Mi , V (2) there exists a finite meet decomposition A = i<n Mi , n < ω, (3) the Boolean algebra Reg(A) is finite? In the next section, we find a connection between this problem and existence of independent bases for anti-identities. 6. Anti-identities of finite structures Recall that a set Σ of anti-identities is a basis for anti-identities of a class K if K is the class of structures in which all anti-identities of Σ are valid, i.e., K = Mod(Σ). By a basis for anti-identities of a structure A we mean a basis for anti-identities of the antivariety generated by A. A basis Σ is said to be independent if, for every ϕ ∈ Σ, the proper inclusion Mod(Σ) ( Mod(Σ \ {ϕ}) holds. 136 A. Kravchenko A structure A is said to be weakly atomic compact if every locally consistent in A set of atomic formulas is consistent in A. We reduce the question on existence of an independent basis for antiidentities of a finite relation structure of finite signature to Problem 14. Let A be a finite relation structure of finite signature and let Σ = (ϕ i )i∈I be an independent basis for anti-identities of A. With each anti-identity ϕ i , i ∈ I, we associate a finitely presented structure B i as follows: Let ϕi ⇌ ∀x(¬ ψ1 (x) ∨ · · · ∨ ¬ ψn (x)); then Bi is the structure defined by generators x and relations ψ1 (x), . . . , ψn (x). It is easy to see that the antivariety defined by Σ coincides with the class T [Bi 9] = {B : Bi 9 B for all i ∈ I}. i∈I Since Σ is an independent basis, we have B i → Bj if and only if i = j. Lemma 15. The following two conditions are equivalent: (1) [→ A] = T i∈I [Bi 9], (2) there exists a family of finite structures Q (A i )i∈I such that [Bi 9] = [→ Ai ] for all i ∈ I and [→ A] = [→ i∈I Ai ]. P roof. It is clear that (2) implies (1). Indeed, if such a family exists then, for every structure C, we have   Q C ∈ [→ A] = → Ai ⇐⇒ C → Ai for all i ∈ I ⇐⇒ i∈I ⇐⇒ C ∈ [Bi 9] for all i ∈ I ⇐⇒ ⇐⇒ C ∈ T [Bi 9] . i∈I We prove that (1) implies (2). Notice that each structure Bi , i ∈ I, is connected. Assume the contrary, i.e., let there exist an element i ∈ I such that B i is not connected. Then Bi = B1i +B2i for some structures Bki with Bi 9 Bki , k = 1, 2. Since Bj 9 Bi k provided T j 6= i, we have Bi ∈ [Bj 9] for all j 6= i and k = 1, 2. Therefore, k Bi ∈ i∈I [Bi 9] = [→ A], k = 1, 2. Thus, Bi ∈ [→ A] ⊆ [Bi 9], which is a contradiction. Lattices of relative colour-families and antivarieties 137 For an arbitrary i ∈ I, consider the interval [A, A + B i ] of the partially ordered set of cores. If there exists a core T C such that A → C → A + B i and Bi 9 C 9 A then, by (1), we obtain C ∈ / i∈I [Bi 9]. Hence, there exists a j ∈ I such that Bj → C. It is easy to see that i 6= j. Since Bj → C → A + Bi and Bj is connected, we conclude that Bj → Bi , where i 6= j. Since Σ is an independent basis, we arrive at a contradiction. Thus, A + B i covers A in the partially ordered set of cores (in symbols: A ≺ A + B i ). Since this is a distributive lattice (cf. Remark 2), we conclude that A×B i ≺ Bi . We denote i Ci = A × Bi , i ∈ I. By (2) and (3), we obtain Ci h→ A →i CB i for all i ∈ I. i = [Bi 9] holds. By [5, Lemma 2.5], for every i ∈ I, the equality → CB i   T Q Bi Put Ai = Ci . We prove that i∈I [Bi 9] = → i∈I Ai . We deduce D∈ T [Bi 9] ⇐⇒ D ∈ [Bi 9] for all i ∈ I ⇐⇒ i∈I ⇐⇒ D ∈ [→ Ai ] for all i ∈ I ⇐⇒  ⇐⇒ D ∈ → Q i∈I  Ai .  Q  T Since i∈I [Bi 9] = [→ A], we obtain [→ A] = → i∈I Ai . Moreover, if Q the structure A has no trivial substructure then the structure i∈I Ai has no trivial substructure either. In the sequel, we assume that (equivalent) conditions (1) and (2) of Lemma 15 are satisfied. Without loss of generality, we may assume that A is a core. Since A is finite,  2.2],  Proposition Q the class [→ A] is elementary. By [2, Q the structure i∈I Ai is Q weakly atomic compact and → i∈I Ai is the antivariety generated by i∈I Ai . In view Q of [2, Corollary 2.6], there exists ∗ of a unique (up to isomorphism) core A i∈I Ai ; moreover, the antivarieties Q ∗ generated by A and i∈I Ai coincide. Therefore, the antivarieties generated by A and A∗ coincide. By [2, Corollary 2.5], the structures A and A ∗ are isomorphic. Since A∗ is a finite structure of finite signature, there exists a finite Q ∗ subset F ⊆ I such that A is embeddable into i∈F Ai . We suggest the following 138 A. Kravchenko   Q Conjecture 16. The equality [→ A] = → i∈F ′ Ai holds for some finite subset F ′ ⊆ I with F ⊆ F ′ . If this conjecture is true then Σ is a finite basis, which means that every finite relation structure of finite signature having no finite basis for its anti-identities possesses no independent basis for its anti-identities. We return to Problem 14. Let Ki be the principal colour-family generated by Ai , i ∈ I. Then [→ A] = V Ki i∈I is an irredundant meet decomposition of [→ A] (in the lattice of colourfamilies). 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