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Symmetric implication zroupoids and weak associative laws

  • Juan M. Cornejo
  • Hanamantagouda P. SankappanavarEmail author
Foundations
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Abstract

An algebra \({\mathbf {A}} = \langle A, \rightarrow , 0 \rangle \), where \(\rightarrow \) is binary and 0 is a constant, is called an implication zroupoid (\({\mathcal {I}}\)-zroupoid, for short) if \({\mathbf {A}}\) satisfies the identities: \((x \rightarrow y) \rightarrow z \approx ((z' \rightarrow x) \rightarrow (y \rightarrow z)')'\) and \( 0'' \approx 0\), where \(x' := x \rightarrow 0\). An implication zroupoid is symmetric if it satisfies: \(x'' \approx x\) and \((x \rightarrow y')' \approx (y \rightarrow x')'\). The variety of symmetric \({\mathcal {I}}\)-zroupoids is denoted by \({{\mathcal {S}}}\). We began a systematic analysis of weak associative laws (or identities) of length \(\le 4\) in Cornejo and Sankappanavar (Soft Comput 22(13):4319–4333, 2018a.  https://doi.org/10.1007/s00500-017-2869-z), by examining the identities of Bol–Moufang type, in the context of the variety \({{\mathcal {S}}}\). In this paper, we complete the analysis by investigating the rest of the weak associative laws of length \(\le 4\) relative to \({{\mathcal {S}}}\). We show that, of the (possible) 155 subvarieties of \({{\mathcal {S}}}\) defined by the weak associative laws of length \(\le 4\), there are exactly 6 distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of \({{\mathcal {S}}}\) defined by weak associative laws of length \(\le 4\).

Keywords

Symmetric implication zroupoid Weak associative law Identity of Bol–Moufang type Semilattice with least element 0 

Notes

Acknowledgements

The first author wants to thank the institutional support of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas) and Universidad Nacional del Sur. Hanamantagouda P. Sankappanavar did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors. The authors wish to express their indebtedness to the two anonymous referees for their careful reading of the paper and for their useful suggestions that helped improve the final presentation of this paper.

Compliance with ethical standards

Conflict of interest

Both authors declare that they have no conflict of interests.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Juan M. Cornejo
    • 1
    • 2
  • Hanamantagouda P. Sankappanavar
    • 3
    Email author
  1. 1.Departamento de MatemáticaUniversidad Nacional del SurBahía BlancaArgentina
  2. 2.INMABB - CONICETBuenos AiresArgentina
  3. 3.Department of MathematicsState University of New YorkNew PaltzUSA

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