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Studia Logica

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

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 

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References

  1. 1.
    Balbes, R., and P. H. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, 1974.Google Scholar
  2. 2.
    Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, Springer, New York, 1981. The free, corrected version (2012) is available online as a PDF file at math.uwaterloo.ca/~snburris. It is also available for a free download at Sankappanavar’s profile page at www.researchgate.net.
  3. 3.
    Cornejo, J. M., and H. P. Sankappanavar, Implication zroupoids I, Algebra Universalis, in press, 2015.Google Scholar
  4. 4.
    Cornejo, J. M., and H. P. Sankappanavar, Implication zroupoids II, in preparation.Google Scholar
  5. 5.
    Cornejo, J. M., and H. P. Sankappanavar, Semisimple varieties of implication zroupoids, Soft Computing, in press, 2015.Google Scholar
  6. 6.
    McCune, W., Prover9 and Mace4, http://www.cs.unm.edu/mccune/prover9/.
  7. 7.
    Sankappanavar H.P.: De Morgan algebras: new perspectives and applications. Scientia Mathematica Japonica 75(1), 21–50 (2012)Google Scholar

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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