Products in a Category with One Object

TL;DR

This paper explores monoids with an additional operation modeling endomorphisms of an object satisfying $X=X\times X$, constructing a universal monoid and analyzing its combinatorial properties, with implications for monoid actions.

Contribution

It introduces a universal monoid structure for endomorphisms of objects with $X=X\times X$, revealing new combinatorial insights and action properties.

Findings

If $X$ has a nontrivial endomorphism and $X=X\times X$, then every finite monoid acts faithfully on $X$.

The constructed monoid exhibits a rich combinatorial structure.

The work connects monoid theory with categorical properties of objects.

Abstract

We study monoids equipped with a second binary operation that captures the structure of the endomorphisms of an object $X$ such that $X=X\times X$. We construct a universal monoid of this type and examine some of its rich combinatorial structure. We show that if $X$ has a nontrivial endomorphism and $X=X\times X$, then every finite monoid has a faithful action on $X$.