# Amalgamation property in quasi modal algebras

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(2) 42. SERGIO ARTURO CELANI. (1) ∇ (a ∨ b) = ∇a ∩ ∇b, and (2) ∇0 = A, for all a, b ∈ A (see [2]). The class of qm-algebras is denoted by QMA. Let A be a qm-algebra. For each P ∈ Ul (A) we define the set ∆−1 (P ) = {a ∈ A | ∆a ∩ P 6= ∅} . Lemma 2. [2] Let A ∈ QMA. (1) For each P ∈ Ul (A), ∆−1 (P ) ∈ F i (A) , (2) a ∈ ∆−1 (P ) iff for all Q ∈ Ul (A), if ∆−1 (P ) ⊆ Q then a ∈ Q. Let A ∈ QMA. We define on Ul (A) a binary relation RA by (P, Q) ∈ RA. ⇔ ∀a ∈ A : if ∆a ∩ P 6= ∅, then a ∈ Q ⇔ ∆−1 (P ) ⊆ Q.. Throughout this paper, we will frequently work with Boolean subalgebras of a given Boolean algebra. In order to avoid any confusion, if A ∈ QMA, then we will use the symbol ∆A to denote the corresponding operation of A. A function h : A → B is an homomorphism of quasi-modal algebras, or a q-homomorphism, if h is an homomorphism of Boolean algebras, and IB (h (∆A a)) = ∆B (h (a)) , for any a ∈ A. A quasi-isomorphism is a Boolean isomorphism that is a q-homomorphism. Let A ∈ QMA. Let us consider the relational structure F (A) = hUl (A) , RA i . From the result given in [2] it follows that the Boolean algebra P(Ul(A)) endowed with the operator ∆RA : P(Ul(A)) → Id (P(Ul(A))) defined by ∆RA (O) = I (∆RA (O)) = {U ∈ P(Ul(A)) | U ⊆ ∆RA (O)} , is a quasi-modal algebra. Moreover, the map β : A → P(Ul(A)) defined by β (a) = {P ∈ Ul(A) | a ∈ P }, for each a ∈ A, is a q-homomorphism, i.e. IP(Ul(A)) (β(∆a)) = ∆RA (β(a)), for each a ∈ A. S ∆x) and the Let A ∈ QMA. Let X ⊆ A. Define the ideal ∆X = I( x∈X S 0 filter ∇X = F x∈X ∇x . For a ∈ A we define recursively ∆ a = I (a) , and n+1 n ∆ a = ∆ (∆ a) . Theorem 3. [2] Let A ∈ QMA. Then for all a ∈ A, the following equivalences hold: 1. ∆a ⊆ I (a) ⇔ RA is reflexive. 2 2. ∆a ⊆ ∆2 a ⇔ RA ⊆ RA , i.e., RA is transitive. T ∆x ⇔ RA is symmetrical. 3. I (a) ⊆ ∆∇a = x∈∇a. Rev. Un. Mat. Argentina, Vol 50-1.

(3) AMALGAMATION PROPERTY IN QUASI–MODAL ALGEBRAS. 43. Let A ∈ QMA. We shall say that A is a topological quasi-modal algebra if for every a ∈ A, ∆a ⊆ I (a) and ∆a ⊆ ∆2 a. We shall say that A is a monadic quasi-modal algebra if it is a quasi-topological algebra and I (a) ⊆ ∆∇a for every a ∈ A. From the previous Theorem we get that a quasi-modal algebra A is quasi-topological algebra iff the relation RA is reflexive and transitive. Similarly, a quasi-modal algebra A is a monadic quasi-modal algebra iff the relation RA is an equivalence. Let A and B be two qm-algebras. We shall say that the structure B = hB, ∨, ∧, ∆B , 0, 1i is a quasi-modal subalgebra of A, or qm-subalgebra for short, if B is a Boolean subalgebra of A, and for any a ∈ B, IA (∆B (a)) = ∆A (a) . The following result is given in [1] for Quasi-modal lattices. Lemma 4. Let A be a qm-algebra. Let B be a Boolean subalgebra of A. Then the following conditions are equivalent (1) There is a quasi-modal operator, ∆B : B → Id (B) , such that B = hB, ∨, ∧, ∆B , 0, 1i is a qm-subalgebra of A. (2) For any a ∈ B, ∆B (a) is defined to be IA (∆A (a) ∩ B). Remark 5. Let A be a qm-algebra. Let B be a Boolean subalgebra of A. If there is a quasi-modal operator ∆B : B → Id (B) , such that B = hB, ∨, ∧, ∆B , 0, 1i is a qm-subalgebra of A, then ∆B is unique. The proof of this fact is as follows: First, we note that if H is an ideal of B, and G is a filter of B such that H ∩ G = ∅, then IA (H) ∩ FA (G) = ∅. We suppose now that there exist two quasi-modal operators ∆1 and ∆2 in B such that hB, ∨, ∧, ∆1 , 0, 1i and hB, ∨, ∧, ∆2 , 0, 1i are two qmsubalgebras of A. Then ∆A (a) = IA (∆1 (a)) = IA (∆2 (a)) . If ∆1 (a) ∆2 (a), there exists b ∈ B such that b ∈ ∆1 (a) and b ∈ / ∆2 (a). Then ∆A (a)∩FA (b) = IA (∆1 (a))∩FA (b) 6= ∅ and ∆A (a)∩FA (b) = IA (∆2 (a))∩FA (b) = ∅, which is a contradiction. 2. Amalgamation Property It is known that the variety of modal algebras has the Amalgamation Property (AP) and the Superamalgamation Property (SAP) (see [3] for these properties and the connection with the Interpolation property in modal logic). In this section we shall give a generalization of these notions and prove that the class QMA has these properties. Definition 6. Let Q be a class of quasi-modal algebras. We shall say that Q has the AP if for any triple A0 , A1 , A2 ∈ Q and injective quasi-homomorphisms f1 : A0 → A1 and f2 : A0 → A2 there exists B ∈ Q and injective quasi-homomorphisms g1 : A1 → B and g2 : A2 → B such that g1 ◦f1 = g2 ◦f2 , and g1 (∆A0 c) = g2 (∆A0 c), for every c ∈ A0 . Rev. Un. Mat. Argentina, Vol 50-1.

(4) 44. SERGIO ARTURO CELANI. We shall say that Q has the SAP if Q has the AP and in addition the maps g1 and g2 above have the following property: For all (a, b) ∈ A1 × A2 such that g1 (a) ≤ g2 (b) there exists c ∈ A0 such that a ≤ f1 (c) and f2 (c) ≤ b. Let Q be a class of quasi-modal algebras. Without losing generality, we can assume that in the above definition A0 is a qm-subalgebra of A1 and A2 , i.e., that f1 and f2 are the inclusion maps. Theorem 7. The class QMA has the AP. Proof. Let A0 , A1 , A2 ∈ QMA such A0 is a qm-subalgebra of A1 and A2 . Let us consider the relational structures F (Ai ) = hUl (Ai ) , RAi , β (Ai )i , 1 ≤ i ≤ 2. We shall define the set X = {(P, Q) ∈ Ul (A1 ) × Ul (A2 ) | P ∩ A0 = Q ∩ A0 }, and the binary relation R in X as follows: (P1 , Q1 ) R (P2 , Q2 ) iff (P1 , P2 ) ∈ RA1 and (Q1 , Q2 ) ∈ RA2 .. ¯ R , ∅ where the Let us consider the quasi-modal algebra A (X) = P (X) , ∪,c , ∆ operator ¯ R : P (X) → Id (P (X)) ∆ is defined by ¯ R (G) = I (∆R (G)) = {U ∈ P (X) | U ⊆ ∆R (G)} . ∆ We note that in this case I (∆R ) = ∆R . Let us define the maps g1 : A1 → A (X) and g2 : A2 → A (X) by g1 (a) = {(P, Q) ∈ X : a ∈ P } and g1 (b) = {(P, Q) ∈ X | b ∈ Q}, respectively. We prove that g1 and g2 are injective q-homomorpshims. First of all let us check the following property: ∀P ∈ Ul (A1 ) ∃Q ∈ Ul (A2 ) such that (P, Q) ∈ X.. (1). Let P ∈ Ul (A1 ) . Since P ∩ A0 ∈ Ul (A0 ) and A0 is a subalgebra of A2 , we get by known results on Boolean algebras that there exists Q ∈ Ul (A2 ) such that P ∩ A0 = Q ∩ A0 , i.e., (P, Q) ∈ X. To see that g1 is injective, let a, b ∈ A1 such that a b. Then there exists P ∈ Ul (A1 ) such that a ∈ P and b ∈ / P. By (1), there exists Q ∈ Ul (A2 ) such that (P, Q) ∈ X. So, (P, Q) ∈ g1 (a) and (P, Q) ∈ / g1 (b) , i.e., g1 (a) g1 (b) . Thus, g1 is injective. It is clear that g1 is a Boolean homomorphism. We prove next that ∆R (g1 (a)) ⊆ g1 (∆a), for any a ∈ A1 . Let (P, Q) ∈ X. Suppose that (P, Q) ∈ / g1 (∆A1 a) , i.e., ∆A1 a ∩ P = ∅. From Lemma 2 there exists P1 ∈ Ul (A1 ) such that (P, P1 ) ∈ RA1 and a ∈ / P1 . By property (1), there exists Q1 ∈ Ul (A2 ) such that P1 ∩ A0 = Q1 ∩ A0 . From the −1 inclusion ∆−1 A1 (P ) ⊆ P1 , it is easy to see that ∆A0 (P ∩ A0 ) ⊆ P1 ∩ A0 . Thus, ∆−1 A0 (P ∩ A0 ) ⊆ Q1 ∩ A0 . Rev. Un. Mat. Argentina, Vol 50-1.

(5) AMALGAMATION PROPERTY IN QUASI–MODAL ALGEBRAS. 45. We prove that there exists Q2 ∈ Ul (A2 ) such that P1 ∩A0 = Q2 ∩A0 and ∆−1 A2 (Q) ⊆ Q2 . Let us consider the filter F = FA2 ∆−1 A2 (Q) ∪ (P1 ∩ A0 ) . −1 The filter F is proper, because otherwise there exist q ∈ ∆A (Q) and d ∈ P1 ∩ A0 2 such that q ∧ d = 0. So, q ≤ ¬d, and since ∆A2 is increasing, we get ∆A2 (q) ⊆ ∆A2 (¬d) . As A0 is a quasi-subalgebra of A2 and d ∈ A0 , we get. ∆A0 (¬d) = IA2 (∆A2 (¬d) ∩ A0 ) . Thus, IA2 (∆A2 (¬d) ∩ A0 ) ∩ Q 6= ∅. So, there exists x ∈ Q and y ∈ ∆A2 (¬d) ∩ A0 such that x ≤ y. Since y ∈ Q ∩ A0 = P ∩ A0 , y ∈ ∆A2 (¬d) ∩ A0 ∩ P. Then we have ¬d ∈ P1 , which is a contradiction. Therefore, there exists Q2 ∈ Ul (A2 ) such that P1 ∩ A0 = Q2 ∩ A0 and ∆−1 A2 (Q) ⊆ Q2 . So, (P, Q) ∈ / ∆R g1 (a) . So, we have the inclusion ∆R g1 (a) ⊆ g1 (∆A1 a). The inclusion g1 (∆A1 a) ⊆ ∆R g1 (a) is easy and left to the reader. Thus, g1 (∆A1 a) = ∆R g1 (a), and consequently we get I (g1 (∆A1 a)) = I (∆R g1 (a)) , i.e., g1 is an injective q-homomorphism. Similarly we can prove that g2 is an injective q-homomorphism. Moreover, it is easy to check that for all c ∈ A0 , g1 (c) = g2 (c), and g1 (∆A0 c) = g2 (∆A0 c). Thus, QMA has the AP. Lemma 8. Let A0 , A1 and A2 ∈ QMA such that A0 is a subalgebra of A1 and A2 . Let a ∈ A1 , b ∈ A2 and let us suppose that there exists no c ∈ A0 such that a ≤ c or c ≤ b. Then there exist P ∈ Ul (A1 ) , Q ∈ Ul (A2 ) such that a ∈ P, b ∈ / Q and P ∩ A0 = Q ∩ A0 . Proof. Let us consider the filter FA1 (a) in A1 and the filter FA2 (FA1 (a) ∩ A0 ) in A2 . We note that b∈ / FA2 (FA1 (a) ∩ A0 ) , because in otherwise there exists c ∈ A0 such that a ≤1 c ≤2 b, which is a contradiction. Then by the Ultrafilter theorem, there exists Q ∈ Ul (A2 ) that such that / Q. FA1 (a) ∩ A0 ⊂ FA2 (FA1 (a) ∩ A0 ) ⊆ Q and b ∈ Let us consider in A1 the filter F = FA1 ({a} ∪ (Q ∩ A0 )) and the ideal I = IA1 ((A2 − Q) ∩ A0 ). We prove that F ∩ I = ∅. Suppose that there exist elements x ∈ A1 , y ∈ Q ∩ A0 , and z ∈ (A2 − Q) ∩ A0 such that a ∧ y ≤1 x ≤1 z. This implies that a ≤1 ¬y ∨ z ∈ FA1 (a) ∩ A0 ⊆ Q. Rev. Un. Mat. Argentina, Vol 50-1.

(6) 46. SERGIO ARTURO CELANI. Thus, we get z ∈ Q, which is a contradiction. So, there exists P ∈ Ul (A1 ) such that a ∈ P, Q ∩ A0 = P ∩ A0 and b ∈ / Q. Theorem 9. The class QMA has the SAP. Proof. The proof of the SAP is actually analogous to the previous one. Let A0 , A1 , A2 ∈ QMA such that A0 is a qm-subalgebra of A1 and of A2 . Let us consider the set X and the quasi-modal algebra A (X) of the proof above. Let (a, b) ∈ A1 × A2 . Suppose that there exists no c ∈ A0 such that a ≤ c or c ≤ b. By Lemma 8 there exists P ∈ Ul (A1 ) and there exists Q ∈ Ul (A2 ) such that a ∈ P, b ∈ / Q and P ∩ A0 = Q ∩ A0 , i.e., (P, Q) ∈ g1 (a) and (P, Q) ∈ / g2 (a) . Thus, g1 (a) * g2 (b) . So, QMA has the SAP. The results above can be applied to prove that other classes of quasi-modal algebras have the AP and SAP. For example, if T QMA is the class of the topological quasi-modal algebras, then in the proof of Theorem 7 the binary relation R defined on the set X is reflexive and transitive. Consequently, A (X) is a topological quasi-modal algebra and thus the class T QMA has the AP and the SAP. Similar considerations can be applied to the class MQMA of monadic quasi-modal algebras. Acknowledgement I would like to thank the referee for his observations and suggestions which have contributed to improve this paper. References [1] Castro, J. and Celani, S., Quasi-modal lattices, Order 21 (2004), 107-129. [2] Celani, S. A., Quasi-Modal algebras, Mathematica Bohemica Vol. 126, No. 4 (2001), 721-736. [3] Kracht. M., Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of Mathematics, Vol. 142, Elsevier.. Sergio Arturo Celani CONICET and Departamento de Matem´ aticas Universidad Nacional del Centro Pinto 399, 7000 Tandil, Argentina scelani@exa.unicen.edu.ar. Recibido: 9 de marzo de 2007 Aceptado: 19 de septiembre de 2008. Rev. Un. Mat. Argentina, Vol 50-1.

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