Soft Computing

, Volume 21, Issue 23, pp 6963–6982

# On derived algebras and subvarieties of implication zroupoids

• Juan M. Cornejo
• Hanamantagouda P. Sankappanavar
Foundations

## Abstract

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:; and Soft Comput: 20:3139–3151, 2016c. doi:), 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).

## Keywords

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

## Notes

### Acknowledgements

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.

### 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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