Soft Computing

, Volume 22, Issue 13, pp 4319–4333 | Cite as

Symmetric implication zroupoids and identities of Bol–Moufang type

  • Juan M. Cornejo
  • Hanamantagouda P. SankappanavarEmail author


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: (I): \((x \rightarrow y) {\rightarrow } z \approx ((z' {\rightarrow } x) {\rightarrow } (y {\rightarrow } z)')'\), and (I\(_{0}\)): \( 0'' \approx 0\), where \(x' : = x \rightarrow 0\). An implication zroupoid is symmetric if it satisfies the identities: \(x'' \approx x\) and \((x \rightarrow y')' \approx (y \rightarrow x')'\). An identity is of Bol–Moufang type if it contains only one binary operation symbol, one of its three variables occurs twice on each side, each of the other two variables occurs once on each side, and the variables occur in the same (alphabetical) order on both sides of the identity. In this paper, we will present a systematic analysis of all 60 identities of Bol–Moufang type in the variety \({\mathcal {S}}\) of symmetric \({\mathcal {I}}\)-zroupoids. We show that 47 of the subvarieties of \({\mathcal {S}}\), defined by the identities of Bol–Moufang type, are equal to the variety \({\mathcal {SL}}\) of \(\vee \)-semilattices with the least element 0 and one of others is equal to \({\mathcal {S}}\). Of the remaining 12, there are only three distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of \({\mathcal {S}}\) of Bol–Moufang type.



The first author wants to thank the institutional support of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas). The authors wish to express their indebtedness to the anonymous referees for their useful suggestions that helped improve the final presentation of this paper. The work of Juan M. Cornejo was supported by CONICET (Consejo Nacional de Investigaciones Cientificas y Tecnicas) 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.

Compliance with ethical standards

Conflict of interest

The first author declares that he has no conflict of interest. The second author declares that he has no conflict of interest.

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 2017

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 - CONICETBahía BlancaArgentina
  3. 3.Department of MathematicsState University of New YorkNew PaltzUSA

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