Symmetric Implication Zroupoids and Weak Associative Laws

JUAN MANUEL CORNEJO; HANAMANTAGOUDA P. SANKAPPANAVAR

SOFT COMPUTING - (Print)

SPRINGER

Lugar: Berlin; Año: 2019 vol. 23 p. 6797 - 6812

1472-7643

An algebra $mathbf A = langle A, o, 0 angle$, where $o$ is binary and $0$ is a constant, is called an {it implication zroupoid} ($mathcal I$-zroupoid, for short) if $mathbf A$ satisfies the identities: $(x o y) o z approx ((z´ o x) o (y o z)´)´$ and $ 0´´ approx 0$, where $x´ : = x o 0$. An implication zroupoid is {it symmetric} if it satisfies: $x´´ approx x$ and $(x o y´)´ approx (y o x´)´$. 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 $leq 4$ in cite{cornejo2016BolMoufang}, 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 $leq 4$ relative to $mathcal S$. We show that, of the (possible) 155 subvarieties of $mathcal S$ defined by the weak associative laws of length $leq 4$, 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 $leq 4$.