On the local k-elasticities of Puiseux monoids

Abstract

If [Formula: see text] is an atomic monoid and [Formula: see text] is a nonzero non-unit element of [Formula: see text], then the set of lengths [Formula: see text] of [Formula: see text] is the set of all possible lengths of factorizations of [Formula: see text], where the length of a factorization is the number of irreducible factors (counting repetitions). In a recent paper, F. Gotti and C. O’Neil studied the sets of elasticities [Formula: see text] of Puiseux monoids [Formula: see text]. Here, we take this study a step further and explore the local [Formula: see text]-elasticities of the same class of monoids. We find conditions under which Puiseux monoids have all their local elasticities finite as well as conditions under which they have infinite local [Formula: see text]-elasticities for sufficiently large [Formula: see text]. Finally, we focus our study of the [Formula: see text]-elasticities on the class of primary Puiseux monoids, proving that they have finite local [Formula: see text]-elasticities if either they are boundedly generated and do not have any stable atoms or if they do not contain [Formula: see text] as a limit point.