Abstract
The catenary degree of an element [Formula: see text] of a cancellative commutative monoid [Formula: see text] is a nonnegative integer measuring the distance between the irreducible factorizations of [Formula: see text]. The catenary degree of the monoid [Formula: see text], defined as the supremum over all catenary degrees occurring in [Formula: see text], has been studied as an invariant of nonunique factorization. In this paper, we investigate the set [Formula: see text] of catenary degrees achieved by elements of [Formula: see text], focusing on the case where [Formula: see text] is finitely generated (where [Formula: see text] is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of [Formula: see text] that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for [Formula: see text].
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