Studia Logica

, Volume 104, Issue 3, pp 417–453

# Order in Implication Zroupoids

• Juan M. Cornejo
• Hanamantagouda P. Sankappanavar
Article

## Abstract

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

## Keywords

Implication zroupoid Partial order Boolean algebra De Morgan algebra The variety $${I_{2,0}}$$ finite $${I_{2,0}}$$-chain

## References

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