Semisimple Varieties of Implication Zroupoids

TL;DR

This paper characterizes the simple algebras within the variety of implication zroupoids, showing there are exactly five, and describes the structure of their semisimple subvarieties, revealing a specific lattice structure.

Contribution

It identifies all simple algebras in the variety I of implication zroupoids and describes the lattice structure of their semisimple subvarieties.

Findings

Exactly five simple algebras in I.

Semisimple subvarieties are generated by these five algebras.

Lattice of semisimple subvarieties is isomorphic to a product of a Boolean lattice and a chain.

Abstract

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In 2012, this result was extended to De Morgan algebras in [8] which led Sankappanavar to introduce, and investigate, the variety I of implication zroupoids generalizing De Morgan algebras. His investigations were continued in [3] and [4] in which several new subvarieties of I were introduced and their relationships with each other and with the varieties of [8] were explored. The present paper is a continuation of [8] and [3]. The main purpose of this paper is to determine the simple algebras in I. It is shown that there are exactly five simple algebras in I. From this description we deduce that the semisimple subvarieties of I are precisely the subvarieties of the variety generated by these 5 simple I-zroupoids and are locally finite. It also follows that the lattice of semisimple subvarieties of I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.