Complete Heyting Algebra-Valued Convergence Semigroups

Abstract

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