Embedding groups into distributive subsets of the monoid of binary operations

Abstract

Let $X$ be a set and $Bin\left(X\right)$ the set of all binary operations on $X$. We say that $S\subset Bin\left(X\right)$ is a distributive set of operations if all pairs of elements ${\ast }_{\alpha },{\ast }_{\beta }\in S$ are right distributive, that is, $\left(a{\ast }_{\alpha }b\right){\ast }_{\beta }c=\left(a{\ast }_{\beta }c\right){\ast }_{\alpha }\left(b{\ast }_{\beta }c\right)$ (we allow ${\ast }_{\alpha }={\ast }_{\beta }$).

The question of which groups can be realized as distributive sets was asked by J. Przytycki. The initial guess that embedding into $Bin\left(X\right)$ for some $X$ holds for any $G$ was complicated by an observation that if $\ast \in S$ is idempotent ($a\ast a=a$), then $\ast$ commutes with every element of $S$. The first noncommutative subgroup of $Bin\left(X\right)$ (the group $S_3$) was found in October 2011 by Y. Berman.

Here we show that any group can be embedded in $Bin\left(X\right)$ for $X=G$ (as a set). We also discuss minimality of embeddings observing, in particular, that $X$ with six elements is the smallest set such that $Bin\left(X\right)$ contains a nonabelian subgroup.