Order in Implication Zroupoids

Abstract

The variety I of implication zroupoids (using a binary operation → and a constant 0) was defined and investigated by Sankappanavar (Scientia Mathematica Japonica 75(1):21–50, 2012), as a generalization of De Morgan algebras. Also, in Sankappanavar (Scientia Mathematica Japonica 75(1):21–50, 2012), several subvarieties of I were introduced, including the subvariety I2 ,0, defined by the identity: x″≈ x, which plays a crucial role in this paper. Some more new subvarieties of I are studied in Cornejo and Sankappanavar (Algebra Univ, 2015) that includes the subvariety SL of semilattices with a least element 0. An explicit description of semisimple subvarieties of I is given in Cornejo and Sankappanavar (Soft Computing, 2015). It is a well known fact that there is a partial order (denote it by ⊑) induced by the operation ∧, both in the variety SL of semilattices with a least element and in the variety DM of De Morgan algebras. As both SL and DM are subvarieties of I and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation ⊑ on I is actually a partial order in some (larger) subvariety of I that includes both SL and DM. The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety I2,0 is a maximal subvariety of I with respect to the property that the relation ⊑ is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in I2,0 that can be defined on an n-element chain (herein called I2,0-chains), n being a natural number. Secondly, we answer this problem in our second main theorem which says that, for each n∈ N, there are exactly n nonisomorphic I2, 0-chains of size n.