On implicator groupoids

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 ${\mathcal{I}}$ of algebras, there called implication zroupoids (I-zroupoids) and here called implicator groupoids (${\mathcal{I}}$-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 ${\mathcal{I}}$, and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of ${\mathcal{I}}$ are introduced and their relationship with each other, and with the subvarieties of ${\mathcal{I}}$ which were already investigated in the paper mentioned above, are explored.