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arXiv · 2311.06901

Ideal extensions of free commutative monoids

C. Cisto·P. A. Garcia-Sanchez·D. Llena
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Abstract

We introduce a new family of monoids, which we call gap absorbing monoids. Every gap absorbing monoid is an ideal extension of a free commutative monoid. For a gap absorbing monoid $S$ we study its set of atoms and Betti elements, which allows us to show that the catenary degree of $S$ is at most four and that the set of lengths of any element in $S$ is an interval. We also give bounds for the $ω$-primality of any ideal extension of a free commutative monoid. For ideal extensions $S$ of $\mathbb{N}^d$, with $d$ a positive integer, we show that $ω(S)$ is finite if and only if $S$ has finitely many gaps.

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C. Cisto, P. A. Garcia-Sanchez, D. Llena. 2023-11-12. Ideal extensions of free commutative monoids. https://arxiv.org/abs/2311.06901

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