Symmetric Implication Zroupoids and Weak Associative Laws

TL;DR

This paper completes a systematic study of weak associative laws of length up to 4 in symmetric implication zroupoids, identifying only 6 distinct subvarieties among 155 possible, and describing their hierarchical structure.

Contribution

It provides a complete classification of subvarieties of symmetric implication zroupoids defined by weak associative laws of length up to 4, expanding previous partial analyses.

Findings

Exactly 6 subvarieties among 155 are defined by these laws.

The poset of these subvarieties is explicitly described.

The analysis completes the classification of weak associative laws in this context.

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 $\leq 4$.