Topological representation for implication algebras

Abstract

In this paper we give a description of an implication algebra A as a union of a unique family of filters of a suitable Boolean algebra Bo(A), called the Boolean closure of A. From this representation we obtain a notion of topological implication space and we give a dual equivalence based in the Stone representation for Boolean algebras. As an application we provide the implication space of all free implication algebras.

Topological representation for monadic implication algebras

“…Abad, J. P. Díaz Varela and A. Torrens gave in [2] a topological representation for implication algebras based on the Stone topological representation for Boolean algebras. We begin by explaining these results briefly.…”

“…It has been proved in [2] that for any implication algebra A, Bo(A) is the least, up to isomorphism, Boolean algebra in which the filter F (A) is an ultrafilter. As two implication algebras may have the same Boolean closure, they are distinguished by means of the family of all maximal elements in the set of all filters of Bo(A) contained in A.…”

Zariski‐type topology for implication algebras

“…In [1] we represent an implication algebra as a union of a unique family of lters of a suitable Boolean algebra Bo(A), and we use the Stone space of Bo(A) to obtain a topological representation for A. Now we de ne a Zariski type topology on the set Spec(A) of maximal implicative lters of A in such a way that the Stone space of Bo(A) is homeomorphic to the one-point compacti cation of the topological space Spec(A).…”

“…Let B(A) be the Boolean subalgebra of B generated by A, and F (A) the lter generated by A in B(A). A is increasing in B(A) [1] (a new shorter proof is given in Lemma 2.2), and consequently, A is a union of lters of B(A).…”

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In this work we provide a new topological representation for implication algebras in such a way that its onepoint compacti cation is the topological space given in [1]. Some applications are given thereof.

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