Order in Implication Zroupoids

Abstract

The variety $${\mathbf{I}}$$I of implication zroupoids (using a binary operation $${\to}$$� 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 $${\mathbf{I}}$$I were introduced, including the subvariety $${\mathbf{I_{2,0}}}$$I2,0, defined by the identity: $${x^{\prime \prime}\approx x}$$x��x, which plays a crucial role in this paper. Some more new subvarieties of $${\mathbf{I}}$$I are studied in Cornejo and Sankappanavar (Algebra Univ, 2015) that includes the subvariety $${\mathbf{SL}}$$SL of semilattices with a least element 0. An explicit description of semisimple subvarieties of $${\mathbf{I}}$$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 $${\sqsubseteq}$$�) induced by the operation �, both in the variety $${\mathbf{SL}}$$SL of semilattices with a least element and in the variety $${\mathbf{DM}}$$DM of De Morgan algebras. As both $${\mathbf{SL}}$$SL and $${\mathbf{DM}}$$DM are subvarieties of $${\mathbf{I}}$$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 $${\sqsubseteq}$$� on $${\mathbf{I}}$$I is actually a partial order in some (larger) subvariety of $${\mathbf{I}}$$I that includes both $${\mathbf{SL}}$$SL and $${\mathbf{DM}}$$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 $${\mathbf{I_{2,0}}}$$I2,0 is a maximal subvariety of $${\mathbf{I}}$$I with respect to the property that the relation $${\sqsubseteq}$$� 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 $${\mathbf{I_{2,0}}}$$I2,0 that can be defined on an n-element chain (herein called $${\mathbf{I_{2,0}}}$$I2,0-chains), n being a natural number. Secondly, we answer this problem in our second main theorem which says that, for each $${n \in \mathbb{N}}$$n�N, there are exactly n nonisomorphic $${\mathbf{I_{2,0}}}$$I2,0-chains of size n.