On Kalman's functor for bounded hemi-implicative semilattices and hemi-implicative lattices

TL;DR

This paper explores the categorical equivalence between bounded hemi-implicative semilattices and hemi-implicative lattices, extending Kalman's classical construction relating bounded distributive lattices and Kleene algebras.

Contribution

It introduces an equivalence of categories between bounded hemi-implicative semilattices and hemi-implicative lattices, inspired by Kalman's construction.

Findings

Establishes categorical equivalence between the two algebraic structures.

Extends Kalman's construction to a broader class of algebraic systems.

Provides a new perspective on the relationship between these lattices.

Abstract

Hemi-implicative semilattices (lattices), originally defined under the name of weak implicative semilattices (lattices), were introduced by the second author of the present paper. A hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,1)$ of type $(2,2,0)$ such that $(H,\wedge)$ is a meet semilattice, $1$ is the greatest element with respect to the order, $a\rightarrow a = 1$ for every $a\in H$ and for every $a$, $b$, $c\in H$, if $a\leq b\rightarrow c$ then $a\wedge b \leq c$. A bounded hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,0,1)$ of type $(2,2,0,0)$ such that $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice and $0$ is the first element with respect to the order. A hemi-implicative lattice is an algebra $(H,\wedge,\vee,\rightarrow,0,1)$ of type $(2,2,2,0,0)$ such that $(H,\wedge,\vee,0,1)$ is a bounded distributive lattice and the reduct algebra $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice. In this paper we introduce an equivalence for the categories of bounded hemi-implicative semilattices and hemi-implicative lattices, respectively, which is motivated by an old construction due J. Kalman that relates bounded distributive lattices and Kleene algebras.