We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Full Nonassociative Lambek Calculus with Distribution DFNL). We prove that formal grammars based on DFNL, also with assumptions, generate context-free languages. The proof uses proof-theoretic tools (interpolation) and a construction of a finite model, employed in [13] in the proof of Strong Finite Mode...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...

. 1 Preliminaries Consider the variety Eω generated by the algebra ω := (N, ·), where i · j := max (i, j) for i 6= j, and i · j := 1 for i = j, i, j ∈ N1. These variety is a subvariety of the variety of equivalential algebras E . By an equivalential algebra we mean a grupoid A = (A,↔) that is a subreduct of a Brouwerian semilattice (or, equivalently, a Heyting algebra) with the operation ↔ give...

In this paper, we will show how one is able to construct a lattice on the completion of an algebra and to obtain an isomorphism to its Caratheodory Extension. In addition, it will be shown that the lattice form a σ-algebra and a complete Heyting algebra of countable type.

We study the proof-theory of co-Heyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a single-assumption multiple-conclusions Natural Deduction system NJ for this logic: unlike the best-known treatments of multiple-conclusion systems (e.g., Parigot’s λ−μ calculus, or Urban and Bierman’s term-calculus) here t...

We show that the variety generated by the three-element Heyting algebra admits a meet dense, regular completion even though it is not closed under MacNeille completions.

A relation on a hypergraph is a binary relation on the set consisting of all the nodes and the edges, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join preserving mappings on the lattice of sub-hypergraphs. This paper introduces a generalization of a relation algebra in which the Boolean algebra part is replaced by a Heyting...

Considering a commutative unital quantale L as the truth value table and using the tool of L-generalized convergence structures of stratified L-filters, this paper introduces a kind of fuzzy upper topology, called fuzzy S-upper topology, on L-preordered sets. It is shown that every fuzzy join-preserving L-subset is open in this topology. When L is a complete Heyting algebra, for every completel...