The category of atomic monoids with atom-preserving homomorphisms is complete and cocomplete, with explicit product, coproduct, equalizer, and pullback constructions and length-set formulas.
On monoid algebras having every nonempty subset of $\mathbb{N}_{\ge 2}$ as a length set
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We construct monoid algebras which satisfy the ascending chain condition on principal ideals and which have the property that every nonempty subset of $\mathbb{N}_{\ge 2}$ occurs as a length set.
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The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties
The category of atomic monoids with atom-preserving homomorphisms is complete and cocomplete, with explicit product, coproduct, equalizer, and pullback constructions and length-set formulas.