Ideal extensions of free commutative monoids

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 $\omega$-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 $\omega(S)$ is finite if and only if $S$ has finitely many gaps.