Abstract

The variety \ of implication zroupoids and a constant 0) was defined and investigated by Sankappanavar :21–50, 2012), as a generalization of De Morgan algebras. Also, in Sankappanavar :21–50, 2012), several subvarieties of \ were introduced, including the subvariety \, defined by the identity: \, which plays a crucial role in this paper. Some more new subvarieties of \ are studied in Cornejo and Sankappanavar that includes the subvariety \ of semilattices with a least element 0. An explicit description of semisimple subvarieties of \ is given in Cornejo and Sankappanavar. It is a well known fact that there is a partial order ) induced by the operation ∧, both in the variety \ of semilattices with a least element and in the variety \ of De Morgan algebras. As both \ and \ are subvarieties of \ 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 \ is actually a partial order in some subvariety of \ that includes both \ and \. 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 \ is a maximal subvariety of \ 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 \ that can be defined on an n-element chain -chains), n being a natural number. Secondly, we answer this problem in our second main theorem which says that, for each \, there are exactly n nonisomorphic \-chains of size n.