This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety $mathbf{RDQDStSH_1}$ of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element $mathbf{RDQDStSH_1}$-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras--...

An algebra M = (M ; ,∨,∧,→, 0, 1) of type 〈2, 2, 2, 2, 0, 0〉 is called a bounded commutative R`-monoid iff (i) (M ; , 1) is a commutative monoid, (ii) (M ;∨,∧, 0, 1) is a bounded lattice, and (iii) x y ≤ z ⇐⇒ x ≤ y → z, (iv) x (x → y) = x ∧ y, for each x, y, z ∈ M . In the sequel, by an R`-monoid we will mean a bounded commutative R`-monoid. (Note that bounded commutative R`-monoids are just bo...

In this paper we study the logic Lωω, which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of Lωω with respect to models in Grothendieck toposes, which can be sharpened t...

A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete archimedean Heyting …eld, a terminal object in the category of archimedean Heyting …elds.

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended $lor$-De Morgan law introduced in cite{Sa12}. Then, using this result and the results of cite{Sa12}, we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) ...

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x∨¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To pr...

Usual normalization by evaluation techniques have a strong relationship with completeness with respect to Kripke structures. But Kripke structures is not the only semantics that ts intuitionistic logic: Heyting algebras are a more algebraic alternative. In this paper, we focus on this less investigated area: how completeness with respect to Heyting algebras generate a normalization algorithm fo...

FLew-algebras form the algebraic semantics of the full Lambek calculus with exchange and weakening. We investigate two relations, called satisfiability and positive satisfiability, between FLew-terms and FLew-algebras. For each FLew-algebra, the sets of its satisfiable and positively satisfiable terms can be viewed as fragments of its existential theory; we identify and investigate the compleme...

Rough set systems induced by equivalences have been proved to exhibit polymorphic logical behaviours in dependence on the extension of the set of completely defined objects. They give a rise to semi-simple Nelson algebras, hence three-valued Lukasiewicz algebras and regular double Stone algebras. Additionally, it has been shown that in the presence of completely defined objects, they fulfil a f...