INMABB   05456
INSTITUTO DE MATEMATICA BAHIA BLANCA
Unidad Ejecutora - UE
artículos
Título:
On derived algebras and subvarieties of implication zroupoids
Autor/es:
JUAN MANUEL CORNEJO; HANAMANTAGOUDA P. SANKAPPANAVAR
Revista:
SOFT COMPUTING - (Print)
Editorial:
SPRINGER
Referencias:
Lugar: Berlin; Año: 2017 vol. 21 p. 6963 - 6982
ISSN:
1472-7643
Resumen:
In 2012, the second author introduced and studied in cite{sankappanavarMorgan2012} the variety $mathcal{I}$ of algebras, called implication zroupoids that generalize De Morgan algebras. An algebra $mathbf A = langle A, o, 0 angle$, where $o$ is binary and $0$ is a constant, is called an emph{implication zroupoid} ($mathcal{I}$-zroupoid, for short) if $mathbf A$ satisfies: $(x o y) o z approx [(z´ o x) o (y o z)´]´$, where $x´ : = x o 0$, and $ 0´´ approx 0$.The present authors devoted cite{cornejo2015implication}, cite{cornejo2016order} and cite{cornejo2016semisimple} 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 $mathbf{A^{m}} := langle A, land, 0 angle$ and $mathbf{A^{mj}} :=langle A, land, lor, 0 angle$ of $mathbf{A} in mathcal{I}$, where $x land y := (x o y´)´$ and $x lor y := (x´ land y´)´$, as well as the lattice of subvarieties of $mathcal{I}$. The varieties $mathcal{I}_{2,0}$,$mathcal{RD}$, $mathcal{SRD}$, $mathcal{C}$, $mathcal{CP}$, $mathcal{A}$, $mathcal{MC}$, and $mathcal{CLD}$ are defined relative to $mathcal{I}$, respectively, by: (I$_{2,0}$) $x´´ approx x$, (RD) $(x o y) o z approx (x o z) o (y o z)$, (SRD) $(x o y) o z approx (z o x) o (y o z)$,(C) $ x o y approx y o x$,(CP) $ x o y´ approx y o x´$,(A) $(x o y) o z approx x o (y o z)$, (MC) $x land y approx y land x$, (CLD) $x o (y o z) approx (x o z) o (y o x)$. The purpose of this paper is two-fold. Firstly, we show that, for each $mathbf A in mathcal{I}$, $mathbf{A}^m$ is a semigroup. From this result, we deduce that, for $mathbf A in mathcal{I}_{2,0} cap mathcal{MC}$, the derived algebra $mathbf{A^{mj}}$ is a distributive bisemilattice and is also a Birkhoff system. Secondly, we show that $mathcal{CLD} subset mathcal{SRD} subset mathcal{RD}$ and $mathcal{C} subset mathcal{CP} cap mathcal{A} cap mathcal{MC} cap mathcal{CLD}$, both of which are much stronger results than were announced in cite{sankappanavarMorgan2012}.