Products in a Category with One Object

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 $X$.

This is a revised version of the paper first posted in April. We have tightened some arguments and added a new section with a conjecture and some further questions. Please send comments to the second author

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• Products in a Category with Only One Object