Symmetric implication zroupoids and identities of Bol-Moufang type

TL;DR

This paper systematically analyzes all 60 Bol-Moufang type identities within symmetric implication zroupoids, revealing that most subvarieties are equivalent to known structures like semilattices, and provides a detailed hierarchy of these subvarieties.

Contribution

It classifies subvarieties of symmetric implication zroupoids defined by Bol-Moufang identities, identifying their equivalences and structure.

Findings

47 subvarieties equal to the variety of semilattices with least element 0

3 distinct subvarieties among the remaining identities

Explicit description of the poset of subvarieties

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: (I): $(x \to y) \to z \approx ((z' \to x) \to (y \to z)')'$, and (I$_{0}$): $ 0'' \approx 0$, where $x' : = x \to 0$. An implication zroupoid is symmetric if it satisfies the identities: $x'' \approx x$ and $(x \to y')' \approx (y \to x')'$. An identity is of Bol-Moufang type if it contains only one binary operation symbol, one of its three variables occurs twice on each side, each of the other two variables occurs once on each side, and the variables occur in the same (alphabetical) order on both sides of the identity. In this paper we make a systematic analysis of all $ 60$ identities of Bol-Moufang type in the variety $\mathcal S$ of symmetric $\mathcal I$-zroupoids. We show that $47$ of the subvarieties of $\mathcal S$, defined by the identities of Bol-Moufang type are equal to the variety $\mathcal{SL}$ of $\lor$-semilattices with the least element $0$ and, one of the others is equal to $\mathcal S$. Of the remaining 12, there are only $3$ distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of $\mathcal S$ of Bol-Moufang type.