Implication semilattice of 990 quasigroup equational laws

TL;DR

This paper classifies 114 equivalence classes and their implications among 990 quasigroup equational laws, exploring their algebraic structure and relation to lattice theory.

Contribution

It systematically determines all equivalence classes and implications among a large set of quasigroup laws, extending Schr"oder's lattice analysis.

Findings

Identified 114 equivalence classes of quasigroup laws.

Mapped all logical implications among these classes.

Included the non-distributive lattice structure within the classification.

Abstract

In his quest to disprove a claim by Peirce that all lattices are distributive, Ernst Schr\"oder considered 135 years ago a list of 990 equational laws on quasigroups, analogous to associativity, such as $(x // y) * z = (y // x) \backslash\backslash z$. A quasigroup is a non-associative analogue of groups, specifically a set equipped with multiplication and right/left conjugate-division operations that are compatible. Each equation of interest identifies two three-variable expressions built from these operations. I determine all $114$ equivalence classes of their conjunctions, and all implications between them. This includes as a small corner the five-element non-distributive lattice identified by Schr\"oder.