Symmetric implication zroupoids and weak associative laws

  • 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: \((x \rightarrow y) \rightarrow z \approx ((z' \rightarrow x) \rightarrow (y \rightarrow z)')'\) 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., 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 \(\le 4\).


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



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.


  1. Balbes R, Dwinger PH (1974) Distributive lattices. University of Missouri Press, ColumbiazbMATHGoogle Scholar
  2. Bernstein BA (1934) A set of four postulates for Boolean algebras in terms of the implicative operation. Trans Am Math Soc 36:876–884MathSciNetzbMATHGoogle Scholar
  3. Burris S, Sankappanavar HP (1981) A course in universal algebra. Springer, New York. The free, corrected version (2012) is available online as a PDF file at
  4. Cornejo JM, Sankappanavar HP (2016a) On implicator groupoids. Algebra Univ 77(2), 125–146. arXiv:1509.03774
  5. Cornejo JM, Sankappanavar HP (2016b) Order in implication zroupoids. Stud Log 104(3):417–453. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cornejo JM, Sankappanavar HP (2016c) Semisimple varieties of implication zroupoids. Soft Comput 20:3139–3151. CrossRefzbMATHGoogle Scholar
  7. Cornejo JM, Sankappanavar HP (2017) On derived algebras and subvarieties of implication zroupoids. Soft Comput 21(23):6963–6982. CrossRefzbMATHGoogle Scholar
  8. Cornejo JM, Sankappanavar HP (2018a) Symmetric implication zroupoids and the identities of Bol–Moufang type. Soft Comput 22(13):4319–4333. CrossRefzbMATHGoogle Scholar
  9. Cornejo JM, Sankappanavar HP (2018b) Implication zroupoids and the identities of associative type. Quasigroups Relat Syst 26:13–34MathSciNetzbMATHGoogle Scholar
  10. Cornejo JM, Sankappanavar HP. Varieties of implication zroupoids I (in preparation) Google Scholar
  11. Fenyves F (1969) Extra loops. II. Publ Math Debr 16:187–192MathSciNetzbMATHGoogle Scholar
  12. Gusev SV, Sankappanavar HP and Vernikov BM (2018) The lattice of varieties of implication semigroups (submitted) Google Scholar
  13. Kunen K (1996) Quasigroups, loops, and associative laws. J Algebra 185:194–204. MathSciNetCrossRefzbMATHGoogle Scholar
  14. McCune W (2005–2010) Prover9 and Mace4.
  15. Phillips JD, Vojtechovsky (2005) The varieties of loops of Bol–Moufang type. Algebra Univ 54:259–271. MathSciNetCrossRefzbMATHGoogle Scholar
  16. Phillips JD, Vojtechovsky (2005) The varieties of quasigroups of Bol–Moufang type: an equational reasoning approach. J Algebra 293:17–33MathSciNetCrossRefzbMATHGoogle Scholar
  17. Rasiowa H (1974) An algebraic approach to non-classical logics. North-Holland, AmsterdamzbMATHGoogle Scholar
  18. Sankappanavar HP (2012) De Morgan algebras: new perspectives and applications. Sci Math Jpn 75(1):21–50MathSciNetzbMATHGoogle Scholar

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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