Products in a Category with One Object
Aaron Gray, Keith Pardue

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 such that . We construct a universal monoid of this type and examine some of its rich combinatorial structure. We show that if has a nontrivial endomorphism and , then every finite monoid has a faithful action on .
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Taxonomy
TopicsRings, Modules, and Algebras · semigroups and automata theory · Advanced Algebra and Logic
