Symmetric Implication Zroupoids and Weak Associative Laws

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: $(x \to y) \to z \approx ((z’ \to x) \to (y \to z)‘)‘$and$0’’ \approx 0$,where$x’ : = x \to 0$. An implication zroupoid is symmetric if it satisfies $x^{\mathrm{\prime }\mathrm{\prime }}\approx x$ and $(x \to y’)’ \approx (y \to x’)'$\mathrm{.}Thevarietyofsymmetric$\mathcal I$-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 $\leq 4$relativeto$\mathcal S$\mathrm{.}Weshowthat,ofthe155subvarietiesof$\mathcal S$definedbytheweakassociativelawsofsize$\leq 4$,thereareexactly$6$ 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 $\le 4$.