Ideal Extensions of Free Commutative Monoids

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 ω-primality of any ideal extension of a free commutative monoid. For ideal extensions S of Nd , with d a positive integer, we show that ω(S) is finite if and only if S has finitely many gaps.