The lattice of varieties of implication semigroups

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.