Weak Implication on Monadic Heyting Algebras

In this paper, we solve an open problem in an special case. The problem is to give a characterization for Heyting algebras by means of fractions. Here, we give a representation for a class of Heyting algebras by means of fractions. Fractions on a bounded distributive lattice is a new algebraic structure, which was recently studied by the authors.

Discussiones Mathematicae - General Algebra and Applications, 2020

In this paper, we initiate the discourse on the properties that hold in an almost semi-Heyting algebra but not in an semi-Heyting almost distributive lattice. We establish an equivalent condition for an almost semi-Heyting algebra to become a Stone almost distributive lattice. Moreover a glance about dense elements in an almost semi-Heyting algebra followed by study of some algebraic properties on them. Finally, we perceive that the kernel of homomorphism is equal to the dense element set.

Pacific Journal of Mathematics, 1985

Mathematical Logic Quarterly, 2020

We provide a recursive construction of all the semi-Heyting algebras that can be defined on a chain with n elements. This construction allows us to count them easily. We also compare the formula for the number of semi-Heyting chains thus obtained to the one previously known.

Demonstratio Mathematica

The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH

Bulletin of the Section of Logic

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. 𝒮ℋ denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras. They share several important properties with Heyting algebras. An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of 𝒮ℋ is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety 𝒮ℋ of asociative type of length 3. Our main result shows that there are 3 such subvarities of 𝒮ℋ.

Categories and General Algebraic Structures With Applications, 2014

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 ∨-De Morgan law introduced in [20]. Then, using this result and the results of [20], we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) in the variety of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. It is shown that there are 25 nontrivial simple algebras in this variety. In Part II, we prove, using the description of simples obtained in this Part, that the variety RDQDStSH1 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 RDQDStSH1-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras-the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of this theorem, we present (equational) axiomatizations for several subvarieties of RDQDStSH1. The Part II concludes with some open problems for further investigation.

Filomat

Considering a complete Heyting algebra H, we introduce a notion of stratified H-convergence semigroup. We develop some basic facts on the subject, besides obtaining conditions under which a stratified H-convergence semigroup is a stratified H-convergence group. We supply a variety of natural examples; and show that every stratified H-convergence semigroup with identity is a stratified H-quasiuniform convergence space. We also show that given a commutative cancellative semigroup equipped with a stratified H-quasi-unifom structure satisfying a certain property gives rise to a stratified H-convergence semigroup via a stratified H-quasi-uniform convergence structure.

Bulletin of the Section of Logic

The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the va...

Logica Universalis, 2013

The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting algebra we associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras.