Studia Logica

, Volume 104, Issue 3, pp 417–453 | Cite as

Order in Implication Zroupoids

  • Juan M. Cornejo
  • Hanamantagouda P. Sankappanavar


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.


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


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Juan M. Cornejo
    • 1
  • Hanamantagouda P. Sankappanavar
    • 2
  1. 1.INMABB - CONICET, Departamento de MatemáticaUniversidad Nacional del SurBahía BlancaArgentina
  2. 2.Department of MathematicsState University of New YorkNew PaltzUSA

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