Ideal Extensions of Free Commutative Monoids

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发表日期：

2025年5月6日

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Carmelo Cisto,
Pedro A. García-Sánchez,
David Llena

摘要

We introduce a new family of monoids, which we call Baeth monoids. Every Baeth monoid is an ideal extension of a free commutative monoid. For a Baeth 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 $\omega $ $ω$ -primality of any ideal extension of a free commutative monoid. For ideal extensions S of ${\mathbb {N}}^d$ $N^{d}$ , with d a positive integer, we show that $\omega (S)$ $ω (S)$ is finite if and only if S has finitely many gaps.