The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (Rl-monoids) are common generalizations of BL-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain i...

In this paper, we  study the Arens regularity properties of module actions. We investigate some properties of topological centers of module actions ${Z}^ell_{B^{**}}(A^{**})$ and  ${Z}^ell_{A^{**}}(B^{**})$ with some conclusions in group algebras.

We expand the notion of characteristic formula to infinite finitely presentable subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presentable subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of infinite algebras ...

By introducing a new operation, the exponentiation of formal languages, we can define Heyting algebras of formal languages. It turns out that some well known families of languages are closed under this exponentiation, e. g., the families of regular and of context-sensitive languages.

The daseinisation is a mapping from an orthomodular lattice in ordinary quantum theory into Heyting algebra topos theory. While distributivity does not always hold lattices, it algebras. We investigate the conditions under which negations and meets are preserved by daseinisation, condition that any element transformed through corresponds to original lattice. show these equivalent, that, only ca...

In this note we prove several duality theorems in lattice theory. We also discuss the connection between spectral spaces and Priestley spaces, and interpret Priestley duality in terms of spectral spaces. The organization of this note is as follows. In the first section we collect appropriate definitions and basic results common to many of the various topics. The next four sections consider Birk...

A new axiomatisation for domain and codomain on semirings and Kleene algebras is proposed. It is much simpler, more general and more flexible than a predecessor, and it is particularly suitable for program analysis and program construction via automated deduction. Different algebras of domain elements for distributive lattices, (co-)Heyting algebras and Boolean algebras arise by adapting this a...

In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.