On some Classes of Heyting Algebras with Successor that have the Amalgamation Property

Abstract

In this paper we shall prove that certain subvarieties of the variety of Salgebras (Heyting algebras with successor) has amalgamation. This result together with an appropriate version of Theorem 1 of [L. L. Maksimova, Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras, Algebra i Logika, 16(6):643-681, 1977] allows us to show interpolation in the calculus IP CS(n), associated with these varieties.We use that every algebra in any of the varieties of S-algebras studied in this work has a canonical extension, to show completeness of the calculus IP C S (n) with respect to appropriate Kripke models.

On the free frontal implicative semilattice extension of a frontal Hilbert algebra

“…However, (X(N ⊕ N 0 )) + is not a Heyting algebra with successor because the (ACC) is not satisfied in the poset X(N ⊕ N 0 ). The fact that (X(N ⊕ N 0 )) + is not a Heyting algebra with successor was also mentioned in [6].…”

On products of posets and coproducts of KM-algebras

“…Moreover, if H is a KM algebra of height n then X(H ) is a poset of height n. If X is a poset of height n, then the set of upsets of X is a KM-algebra of height n (Prop. 2.3 of Castiglioni and San Martín 2011).…”

“…A moment's reflection shows that the Proposition 1.5 of Castiglioni and San Martín (2011) can be given in a more general way: f is a morphism in SE if and only if f is a strict morphism.…”