The lattice of varieties of implication semigroups

TL;DR

This paper fully characterizes the structure of all varieties of implication semigroups, a class of algebras generalizing De Morgan algebras, revealing a non-modular lattice with 16 elements.

Contribution

It provides a complete description of the lattice of all varieties of implication semigroups, including its non-modular structure and element count.

Findings

The lattice of implication semigroup varieties has 16 elements.

The lattice is non-modular.

Complete classification of all varieties within this lattice.

Abstract

In 2012, the second author introduced and examined a new type of algebras as a generalization of De Morgan algebras. These algebras are of type (2,0) with one binary and one nullary operation satisfying two certain specific identities. Such algebras are called implication zroupoids. They invesigated in a number of articles by the second author and J.M.Cornejo. In these articles several varieties of implication zroupoids satisfying the associative law appeared. Implication zroupoids satisfying the associative law are called implication semigroups. Here we completely describe the lattice of all varieties of implication semigroups. It turns out that this lattice is non-modular and consists of 16 elements.