Varieties of Lazy Magmas Characterized by Forbidden Substructure Theorems

TL;DR

This paper characterizes varieties of lazy magmas using forbidden substructure theorems, providing a comprehensive classification of these algebraic structures with computational assistance.

Contribution

It offers the first complete forbidden substructure characterizations of pairs of lazy groupoid varieties, combining theoretical results with computational proof methods.

Findings

Characterization of all pairs of lazy groupoid varieties via forbidden substructures

Some characterizations are straightforward, others are highly involved

Development and sharing of a computational tool for theorem proving in relational algebras

Abstract

A magma (or groupoid) is a set with a binary operation $(A,f)$. Roughly speaking, a magma is said to be lazy if compositions such as $f(x,f(f(y,z),u))$ depend on at most two variables. Recently, Kaprinai, Machida and Waldhauser described the lattice of all the varieties of lazy groupoids. A forbidden structure theorem is one that charcaterizes a smaller class $A$ inside a larger class $B$ as all the elements in $B$ that avoid some substructures. For example, a lattice is distributive (smaller class $A$) if and only if it is a lattice (larger class $B$) and avoids the pentagon and the diamond. In this paper we provide a characterization of all pairs of lazy groupoid varieties $A\le B$ by forbidden substructure theorems. Some of the results are straightforward, but some other are very involved. All of these results and proofs were found using a computational tool that proves theorems of this type (for many different classes of relational algebras) and that we make available to every mathematician.