On the finiteness of certain factorization invariants

Let H be a monoid and πH be the unique extension of the identity map on H to a monoid homomorphism F(H)→H, where we denote by F(X) the free monoid on a set X.Given A⊆H,anA-word z (i.e., an element of F(A)) is minimal if πH(z)=πH(z′) for every permutation z′ of a proper subword of z. The minimal A-elasticity of H is then the supremum of all rational numbers m/n with m,n∈N+ such that there exist minimal A-words a and b of length m and n, resp., with πH(a)=πH(b). Among other things, we show that if H is commutative and A is finite, then the minimal A-elasticity of H is finite. This provides a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where H is cancellative, commutative, and f initely generated modulo units, and A is the set A(H)ofatomsofH. We also demonstrate that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, f initely generated monoid with trivial group of units whose minimal A (H)-elasticity is infinite.