Symmetric Implication Zroupoids and Weak Associative Laws

Abstract

An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid ($\mathcal I$-zroupoid, for short) if $\mathbf A$ satisfies the identities: $(x \to y) \to z \approx ((z' \to x) \to (y \to z)')'$ and $0'' \approx 0$, where $x' : = x \to 0$. An implication zroupoid is symmetric if it satisfies $x'' \approx x$ and $(x \to y')' \approx (y \to x')'$. The variety of symmetric $\mathcal I$-zroupoids is denoted by $\mathcal S$. We began a systematic analysis of weak associative laws of length $\leq 4$ in [CS16e], 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 155 subvarieties of $\mathcal S$ defined by the weak associative laws of size $\leq 4$, 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 $\leq 4$.