Sets of lengths in atomic unit-cancellative finitely presented monoids

Abstract

For an element a of a monoid H, its set of lengths L(a) ⊂ N is the set of all positive integers k for which there is a factorization a = u 1 ·. . .·u k into k atoms. We study the system L(H) = {L(a) | a ∈ H} with a focus on the unions U k (H) ⊂ N which are the unions of all sets of lengths containing a given k ∈ N. The Structure Theorem for Unions -stating that for all sufficiently large k, the sets U k (H) are almost arithmetical progressions with the same difference and global bound -has found much attention for commutative monoids and domains. We show that it holds true for the not necessarily commutative monoids in the title satisfying suitable algebraic finiteness conditions. Furthermore, we give an explicit description of the system of sets of lengths of monoids Bn = a, b | ba = b n for n ∈ N ≥2 . Based on this description, we show that the monoids Bn are not transfer Krull, which implies that their systems L(Bn) are distinct from systems of sets of lengths of commutative Krull monoids and others.2010 Mathematics Subject Classification. 20M13, 20M05, 13A05. Key words and phrases. sets of lengths, unions of sets of lengths, finitely presented monoids, Möbius monoid.

Structural properties of subadditive families with applications to factorization theory

“…We infer from (13) and point (i) that ρ k+m,i − ρ k,i = ρ k+m − ρ k and λ k+m,i − λ k,i = λ k+m − λ k for all large k, which, by (12), is enough to conclude. Theorem 2.15.…”

“…The Structure Theorem for Unions holds for a wealth of cancellative monoids [8,9], and recent work has revealed that the theorem admits a "purely additive" counterpart: This was made possible by the introduction of directed families, and has led, for the first time, to the extension of the theorem to a non-cancellative setting, see [4,Theorem 2.2 and § 3] and [13,Theorem 3.6].…”

A Note on Finitely Generated Modules over a <i>PID</i>

“…Compared to the definition of the set of lengths of elements in monoid in [4], the definition of length of d D ∈ can be given certainly since Lemma 2.2.…”

A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions

“…L(H) is additively closed in certain Krull monoids stemming from module theory ([1, Section 6.C]). Examples in a non-cancellative setting can be found in [20,Theorem 4.5] and a more detailed discussion of the property of being additively closed is given in [18].…”