Semisimple varieties of implication zroupoids

Abstract

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In fact, in 1934, Bernstein (Trans Am Math Soc 36:876---884, 1934) 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 theorem to De Morgan algebras (see Sankappanavar, Sci Math Jpn 75(1):21---50, 2012). Indeed, it is shown in Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) that the varieties of De Morgan algebras, Kleene algebras, and Boolean algebras are term-equivalent, respectively, to the varieties, $$\mathbf {DM}$$DM, $$\mathbf {KL}$$KL, and $$\mathbf {BA}$$BA whose defining axioms use only the implication $$\rightarrow $$ź 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 $$\mathbf {DM}$$DM, $$\mathbf {KL}$$KL and $$\mathbf {BA}$$BA motivated the second author of this paper to introduce, and investigate, the variety $$\mathbf {I}$$I of implication zroupoids, generalizing De Morgan algebras. These investigations are continued by the authors of the present paper in Cornejo and Sankappanavar (Implication zroupoids I, 2015), wherein several new subvarieties of $$\mathbf {I}$$I are introduced and their relationships with each other and with the varieties studied in Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) are explored. The present paper is a continuation of Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) and Cornejo and Sankappanavar (Implication zroupoids I, 2015). The main purpose of this paper is to determine the simple algebras in $$\mathbf {I}$$I. It is shown that there are exactly five (nontrivial) simple algebras in $$\mathbf {I}$$I. From this description we deduce that the semisimple subvarieties of $$\mathbf {I}$$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 $$\mathbf {I}$$I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.