Semisimple Varieties of Implication Zroupoids

HANAMANTAGOUDA P. SANKAPPANAVAR; JUAN MANUEL CORNEJO

SOFT COMPUTING - (Print)

SPRINGER

Año: 2016 vol. 20 p. 3139 - 3151

1472-7643

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In fact, in 1934, Bernstein gave a system of axioms for Boolean algebras in terms of implication only. Though his original axioms were not equational, a quick look at his axioms would reveal that if one adds a constant, then it is not hard to translate his system of axioms into an equational one. Recently, in 2012, the second author of this paper extended this modified Bernstein´s theoremto De Morgan algebras (see cite{sankappanavarMorgan2012}). Indeed, it is shown in cite{sankappanavarMorgan2012} that the varieties of De Morgan algebras, Kleene algebras, and Boolean algebras are term-equivalent, respectively, to the varieties, $mathbf{DM}$, $mathbf{KL}$, and $mathbf{BA}$ whose defining axioms use only the implication $o$ and the constant $0$. The fact that the identity, herein called (I), occurs as one of the two axioms in the definition of each of the varieties $mathbf{DM}$, $mathbf{KL}$ and $mathbf{BA}$ motivated the second author of this paper to introduce, and investigate, the variety $mathbf{I}$ of implication zroupoids, generalizing De Morgan algebras. These investigations are continued by the authors of the present paper in cite{CoSa2015aI}, wherein several new subvarieties of $mathbf{I}$ are introduced and their relationships with each other and with the varieties studied in cite{sankappanavarMorgan2012} are explored. The present paper is a continuation of cite{sankappanavarMorgan2012} and cite{CoSa2015aI}. The main purpose of this paper is to determine the simple algebras in $mathbf{I}$. It is shown that there are exactly five (nontrivial) simple algebras in $mathbf{I}$. From this description we deduce that the semisimple subvarieties of $mathbf{I}$ are precisely the subvarieties of the variety generated by these simple I-zroupoids and that they are locally finite. It also follows that the lattice of semisimple subvarieties of $mathbf{I}$ is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.