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Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009

 

Amalgamation Property in Quasi-Modal algebras

Sergio Arturo Celani

Abstract. In this paper we will give suitable notions of Amalgamation and Super-amalgamation properties for the class of quasi-modal algebras introduced by the author in his paper Quasi-Modal algebras.

2000 Mathematics Subject Classification. 06E25, 03G25.

Key words and phrases. Boolean algebras; Quasi-modal algebras; Amalgamation and Super-amalgamation properties.

1. Introduction and preliminaries

The class of quasi-modal algebras was introduced by the author in [2], as a generalization of the class of modal algebras. A quasi-modal algebra is a Boolean algebra A endowed with a map Δ that sends each element a ∈ A to an ideal Δa of A , and satisfies analogous conditions to the modal operator  of modal algebras [3]. This type of maps, called quasi-modal operators, are not operations on the Boolean algebra, but have some similar properties to modal operators.

It is known that some varieties of modal algebras have the Amalgamation Property (AP) and Superamalgamation Property (SAP). These properties are connected with the Interpolation property in modal logic (see [3]). The aim of this paper is to introduce a generalization of these notions for the class of quasi-modal algebras, topological quasi-modal algebras, and monadic quasi-modal algebras.

We recall some concepts needed for the representation for quasi-modal algebras. For more details see [2] and [1].

Let A = ⟨A, ∨,∧, ¬,0, 1⟩ be a Boolean algebra. The set of all ultrafilters is denoted by Ul (A) . The ideal (filter) generated in A by some subset X ⊆ A, will be denoted by IA(X ), (FA (X )) . The complement of a subset Y ⊆ A will be denoted by Y c or A - Y. The lattice of ideals (filters) of A is denoted by Id(A ) (F i(A) ).

Definition 1. Let A be a Boolean algebra. A quasi-modal operator defined on A is a function Δ : A → Id (A ) that verifies the following conditions for all a, b ∈ A :

  • Q1   Δ (a ∧ b) = Δa ∩ Δb,

  • Q2   Δ1 = A.

A quasi-modal algebra, or qm -algebra, is a pair ⟨A,Δ ⟩ where A is a Boolean algebra and Δ is a quasi-modal operator.

In every quasi-modal algebra we can define the dual operator ∇ : A → F i(A) by ∇a = ¬Δ ¬a, where ¬ Δx = {¬y | y ∈ Δx }. It is easy to see that the operator ∇ satisfies the conditions

  • (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

Δ- 1(P ) = {a ∈ A | Δa ∩ P ⁄= ∅ }.

Lemma 2. [2] Let A ∈ QMA .

  • (1) For each P ∈ Ul (A) ,  - 1 Δ (P) ∈ F i(A ),

  • (2) a ∈ Δ - 1(P) iff for all Q ∈ Ul (A) , if Δ -1 (P ) ⊆ Q then a ∈ Q.

Let A ∈ QMA . We define on Ul (A) a binary relation RA by

(P, Q) ∈ RA ⇔ ∀a-∈1 A : if Δa ∩ P ⁄= ∅, then a ∈ Q ⇔ Δ (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 ΔA 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(ΔAa )) = ΔB (h(a )) ,

for any a ∈ A. A quasi-isomorphism is a Boolean isomorphism that is a q-homomor-phism.

Let A ∈ QMA . Let us consider the relational structure

F (A ) = ⟨Ul(A ),RA ⟩ .

From the result given in [2] it follows that the Boolean algebra P (Ul (A)) endowed with the operator

----- ΔRA : P (Ul(A )) → Id (P (Ul(A )))

defined by

----- ΔRA (O ) = I (ΔRA (O)) = {U ∈ P (Ul (A)) | U ⊆ ΔRA (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(Δa )) = ΔRA (β(a)) , for each a ∈ A .

Let A ∈ QMA. Let X ⊆ A . Define the ideal  ⋃ ΔX = I( Δx ) x∈X and the filter  (⋃ ) ∇X = F x∈X ∇x . For a ∈ A we define recursively Δ0a = I (a), and Δn+1a = Δ (Δna ).

Theorem 3. [2] Let A ∈ QMA . Then for all a ∈ A, the following equivalences hold:

  • 1. Δa ⊆ I (a) ⇔ RA is reflexive.

  • 2. Δa ⊆ Δ2a ⇔ R2A ⊆ RA, i.e., RA is transitive.

  • 3. I (a) ⊆ Δ ∇a = ⋂ Δx ⇔ R x∈∇a A is symmetrical.

Let A ∈ QMA. We shall say that A is a topological quasi-modal algebra if for every a ∈ A, Δa ⊆ I (a) and Δa ⊆ Δ2a. We shall say that A is a monadic quasi-modal algebra if it is a quasi-topological algebra and I (a) ⊆ Δ ∇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 = ⟨B, ∨,∧, ΔB, 0,1⟩ 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 (ΔB (a )) = ΔA (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, Δ : B → Id (B ), B such that B = ⟨B, ∨, ∧,ΔB, 0,1 ⟩ is a qm -subalgebra of A.

  • (2) For any a ∈ B , ΔB (a) is defined to be IA (ΔA (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 ΔB : B → Id (B ), such that B = ⟨B, ∨, ∧,ΔB, 0, 1⟩ is a qm -subalgebra of A, then ΔB 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 Δ 1 and Δ 2 in B such that ⟨B, ∨,∧, Δ1, 0,1⟩ and ⟨B, ∨,∧, Δ2, 0,1⟩ are two qm -subalgebras of A. Then

Δ (a ) = I (Δ (a)) = I (Δ (a )). A A 1 A 2

If Δ1 (a) ⊊ Δ2 (a) , there exists b ∈ B such that b ∈ Δ1 (a) and b ∕∈ Δ2 (a) . Then ΔA (a ) ∩ FA (b) = IA (Δ1 (a)) ∩ FA (b) ⁄= ∅ and Δ (a) ∩ F (b) = I (Δ (a)) ∩ F (b) = ∅, A A A 2 A 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 f : A → A 1 0 1 and f : A → A 2 0 2 there exists B ∈ Q and injective quasi-homomorphisms g1 : A1 → B and g2 : A2 → B such that g1 ∘ f1 = g2 ∘ f2, and g1(ΔA0c ) = g2(ΔA0c ) , for every c ∈ A0 .

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) ∈ A × A 1 2 such that g (a) ≤ g (b) 1 2 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 ) = ⟨Ul (Ai) ,RAi,β (Ai)⟩, 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:

(P ,Q )R (P ,Q ) iff (P ,P ) ∈ R and (Q ,Q ) ∈ R . 1 1 2 2 1 2 A1 1 2 A2

Let us consider the quasi-modal algebra A (X ) = ⟨P (X ) ,∪,c, ¯ΔR, ∅⟩ where the operator

¯ΔR : P (X ) → Id (P (X ))

is defined by

¯ ΔR (G ) = I (ΔR (G)) = {U ∈ P (X ) | U ⊆ ΔR (G )}.

We note that in this case I (ΔR ) = ΔR.

Let us define the maps g : A → A (X ) 1 1 and g : A → A (X ) 2 2 by

g (a) = {(P,Q ) ∈ X : a ∈ P} and g (b) = {(P,Q ) ∈ X | b ∈ Q }, 1 1

respectively. We prove that g 1 and g 2 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 A , 2 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 g 1 is a Boolean homomorphism.

We prove next that ΔR (g1 (a)) ⊆ g1(Δa ) , for any a ∈ A1 .

Let (P,Q ) ∈ X. Suppose that (P, Q) ∕∈ g1 (ΔA1a ), i.e., ΔA1a ∩ 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 inclusion Δ-A 1(P ) ⊆ P1, 1 it is easy to see that Δ -1 (P ∩ A ) ⊆ P ∩ A . A0 0 1 0 Thus,

Δ -A1(P ∩ A0 ) ⊆ Q1 ∩ A0. 0

We prove that there exists Q2 ∈ Ul (A2) such that P1 ∩ A0 = Q2 ∩ A0 and Δ-A 1(Q ) ⊆ Q2. 2 Let us consider the filter

 ( -1 ) F = FA2 Δ A2 (Q ) ∪ (P1 ∩ A0 ) .

The filter F is proper, because otherwise there exist  -1 q ∈ Δ A2 (Q ) and d ∈ P1 ∩ A0 such that q ∧ d = 0 . So, q ≤ ¬d, and since ΔA2 is increasing, we get ΔA2 (q) ⊆ ΔA2 (¬d ). As A0 is a quasi-subalgebra of A2 and d ∈ A0, we get

ΔA0 (¬d ) = IA2 (ΔA2 (¬d) ∩ A0).

Thus,

IA2 (ΔA2 (¬d) ∩ A0) ∩ Q ⁄= ∅.

So, there exists x ∈ Q and y ∈ ΔA2 (¬d ) ∩ A0 such that x ≤ y. Since y ∈ Q ∩ A0 = P ∩ A0, y ∈ ΔA2 (¬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 Δ -A12 (Q ) ⊆ Q2.

So, (P,Q ) ∕∈ ΔRg1 (a) . So, we have the inclusion ΔRg1 (a) ⊆ g1 (ΔA1a ) . The inclusion g1(ΔA1a ) ⊆ ΔRg1 (a) is easy and left to the reader. Thus, g1 (ΔA1a ) = ΔRg1 (a) , and consequently we get

I (g1 (ΔA1a )) = I (ΔRg1 (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(ΔA0c ) = g2(ΔA0c ) . 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 ∈ A 0 such that a ≤ c ≤ b 1 2 , which is a contradiction. Then by the Ultrafilter theorem, there exists Q ∈ Ul(A2 ) that such that

F (a ) ∩ A ⊂ F (F (a) ∩ A ) ⊆ Q and b ∕∈ Q. A1 0 A2 A1 0

Let us consider in A1 the filter F = FA ({a} ∪ (Q ∩ A0 )) 1 and the ideal I = I ((A - Q ) ∩ A ) A1 2 0 . 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 ≤ ¬y ∨ z ∈ F (a ) ∩ A ⊆ Q. 1 A1 0

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) ∈ A × A . 1 2 Suppose that there exists no c ∈ A 0 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 ∩ A = Q ∩ A , 0 0

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.        [ Links ]

[2]    Celani, S. A., Quasi-Modal algebras, Mathematica Bohemica Vol. 126, No. 4 (2001), 721-736.        [ Links ]

[3]    Kracht. M., Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of Mathematics, Vol. 142, Elsevier.         [ Links ]

Sergio Arturo Celani
CONICET and Departamento de Matemáticas
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