Soft Computing

, Volume 21, Issue 23, pp 6963–6982 | Cite as

On derived algebras and subvarieties of implication zroupoids

  • Juan M. Cornejo
  • Hanamantagouda P. Sankappanavar


In 2012, the second author introduced and studied in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012) the variety \({\mathcal {I}}\) of algebras, called implication zroupoids, that generalize De Morgan algebras. 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: \((x \rightarrow y) \rightarrow z \approx [(z' \rightarrow x) \rightarrow (y \rightarrow z)']'\) and \( 0'' \approx 0\), where \(x' : = x \rightarrow 0\). The present authors devoted the papers, Cornejo and Sankappanavar (Alegbra Univers, 2016a; Stud Log 104(3):417–453, 2016b. doi: 10.1007/s11225-015-9646-8; and Soft Comput: 20:3139–3151, 2016c. doi: 10.1007/s00500-015-1950-8), to the investigation of the structure of the lattice of subvarieties of \({\mathcal {I}}\), and to making further contributions to the theory of implication zroupoids. This paper investigates the structure of the derived algebras \(\mathbf {A^{m}} := \langle A, \wedge , 0 \rangle \) and \(\mathbf {A^{mj}} :=\langle A, \wedge , \vee , 0 \rangle \) of \({\mathbf {A}} \in {\mathcal {I}}\), where \(x \wedge y := (x \rightarrow y')'\) and \(x \vee y := (x' \wedge y')'\), as well as the lattice of subvarieties of \({\mathcal {I}}\). The varieties \({\mathcal {I}}_{2,0}\), \({{\mathcal {R}}}{{\mathcal {D}}}\), \(\mathcal {SRD}\), \({\mathcal {C}}\), \({{\mathcal {C}}}{{\mathcal {P}}}\), \({\mathcal {A}}\), \({{\mathcal {M}}}{{\mathcal {C}}}\), and \(\mathcal {CLD}\) are defined relative to \({\mathcal {I}}\), respectively, by: (I\(_{2,0}\)) \(x'' \approx x\), (RD) \((x \rightarrow y) \rightarrow z \approx (x \rightarrow z) \rightarrow (y \rightarrow z)\), (SRD) \((x \rightarrow y) \rightarrow z \approx (z \rightarrow x) \rightarrow (y \rightarrow z)\), (C) \( x \rightarrow y \approx y \rightarrow x\), (CP) \( x \rightarrow y' \approx y \rightarrow x'\), (A) \((x \rightarrow y) \rightarrow z \approx x \rightarrow (y \rightarrow z)\), (MC) \(x \wedge y \approx y \wedge x\), (CLD) \(x \rightarrow (y \rightarrow z) \approx (x \rightarrow z) \rightarrow (y \rightarrow x)\). The purpose of this paper is two-fold. Firstly, we show that, for each \({\mathbf {A}} \in {\mathcal {I}}\), \({\mathbf {A}}^{\mathbf {m}}\) is a semigroup. From this result, we deduce that, for \({\mathbf {A}} \in {\mathcal {I}}_{2,0} \cap {{\mathcal {M}}}{{\mathcal {C}}}\), the derived algebra \(\mathbf {A^{mj}}\) is a distributive bisemilattice and is also a Birkhoff system. Secondly, we show that \(\mathcal {CLD} \subset \mathcal {SRD} \subset {{\mathcal {R}}}{{\mathcal {D}}}\) and \({\mathcal {C}} \subset \ {{\mathcal {C}}}{{\mathcal {P}}} \cap {\mathcal {A}} \cap {{\mathcal {M}}}{{\mathcal {C}}} \cap \mathcal {CLD}\), both of which are much stronger results than were announced in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012).


Implication zroupoid Derived algebras Distributive bisemilattice Birkhoff system Subvarieties Left distributive law Right distributive law Semigroup 



Juan M. Cornejo wants to thank the institutional support of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas). Both authors are grateful to Carina Foresi for helping them with her computer expertise. The authors also wish to express their indebtedness to the anonymous referee for his/her careful reading of an earlier version that helped improve the final presentation of this paper.

Compliance with ethical standards

Conflict of interest

Juan M. Cornejo declares that he has no conflict of interest. Hanamantagouda P. Sankappanavar 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 Berlin Heidelberg 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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