Embedding Groups into Distributive Subsets of the Monoid of Binary Operations

TL;DR

This paper explores how groups can be embedded into the set of binary operations on a set, focusing on distributive subsets and minimal embedding sizes, revealing that any group can be embedded into Bin(X) for X=G.

Contribution

It demonstrates that any group can be embedded into Bin(X) with X set to the group itself, and analyzes the minimal size of sets needed for non-abelian subgroup embeddings.

Findings

Any group can be embedded in Bin(X) for X=G.

Six-element set is minimal for non-abelian subgroup embedding.

Identifies the first noncommutative subgroup in Bin(X).

Abstract

Let X be a set and Bin(X) the set of all binary operations on X. We say that a subset of Bin(X) is a distributive set of operations if all pairs of elements are right distributive. J.Przytycki posed the question of which groups can be realized as distributive sets. The initial guess that any group may be embedded into Bin(X) for some X was complicated by an observation that if a binary operation is idempotent (a*a=a), then it commutes with every element of S. The first noncommutative subgroup of Bin(X), the symmetric group on three elements, was found in October of 2011 by Y.Berman. We show that any group can be embedded in Bin(X) for X=G (as a set). We also discuss minimality of embeddings observing that, in particular, X with six elements is the smallest set such that Bin(X) contains a non-abelian subgroup.