On Implicator Groupoids

TL;DR

This paper explores the structure and relationships of subvarieties within implicator groupoids, extending previous work on their algebraic properties and introducing new subvarieties.

Contribution

It investigates the lattice of subvarieties of implicator groupoids, introduces several new subvarieties, and analyzes their interrelationships, advancing the algebraic theory of these structures.

Findings

Identification of new subvarieties of implicator groupoids

Analysis of the lattice structure of subvarieties

Relationships between new and existing subvarieties

Abstract

In a paper published in 2012, the second author extended the well-known fact that Boolean algebras can be defined using only implication and a constant, to De Morgan algebras-this result led him to introduce, and investigate (in the same paper), the variety I of algebras, there called implication zroupoids (I-zroupoids) and here called implicator gruopids (I- groupoids), that generalize De Morgan algebras. The present paper is a continuation of the paper mentioned above and is devoted to investigating the structure of the lattice of subvarieties of I, and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of I are introduced and their relationship with each other, and with the subvarieties of I which were already investigated in the paper mentioned above, are explored.