Symmetric Implication Zroupoids and Weak Associative Laws

Abstract

An algebra $\mathbf{A}=⟨A,\to ,0⟩$, 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: $(x\to y)\to z\approx ((z^{\prime }\to x)\to (y\to z{)}^{\prime }{)}^{\prime }$ and $0^{\prime \prime }\approx 0$, where $x^{\prime }:=x\to 0$. An implication zroupoid is symmetric if it satisfies $x^{\prime \prime }\approx x$ and $(x\to y^{\prime }{)}^{\prime }\approx (y\to x^{\prime }{)}^{\prime }$. The variety of symmetric $\mathcal{I}$-zroupoids is denoted by $\mathcal{S}$. We began a systematic analysis of weak associative laws of length $\le 4$ 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 $\le 4$ relative to $\mathcal{S}$. We show that, of the 155 subvarieties of $\mathcal{S}$ defined by the weak associative laws of size $\le 4$, 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 $\le 4$.

36 pages