The theory of elementary toposes plays the fundamental role in the categorial analysis of the intuitionistic logic. The main theorem of this theory uses the fact that sets E(A,Ω) (for any object A of a topos E) are Heyting algebras with operations defined in categorial terms. More exactly, subobject classifier true: 1 → Ω permits us define truth-morphism on Ω and operations in E(A,Ω) are define...

In the article we deal with a binary operation that is associative, commutative. We define for such an operation a functor that depends on two more arguments: a finite set of indices and a function indexing elements of the domain of the operation and yields the result of applying the operation to all indexed elements. The definition has a restriction that requires that either the set of indices...

We show that the equational theory of representable lower semilattice-ordered residuated semigroups is finitely based. We survey related results.

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...

Let A and B be lattices with zero. The classical tensor product, A ⊗ B, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: A ⊗ B is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of c...

Let S be a distributive {∨, 0 }-semilattice. In a previous paper, the second author proved the following result: Suppose that S is a lattice. Let K be a lattice, let φ : Conc K → S be a {∨, 0 }-homomorphism. Then φ is, up to isomorphism, of the form Conc f , for a lattice L and a lattice homomorphism f : K → L. In the statement above, Conc K denotes as usual the {∨, 0 }-semilattice of all finit...

We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X . Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X . We introduce the notion of a semidirect product Y ∗ T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X∗ acts doubly on t...