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.
The paper introduces a class of inverse (sub)monoids which contains Jones–Lawson’s gauge inverse (sub)monoid. The aim is to give examples and the basic properties of these monoids. Jones–Lawson’s gauge inverse monoid, as an inverse submonoid of the polycyclic monoid, is the prototype in our development line. The generalization leads also to Meakin–Sapir type results involving bijections between special congruences and special wide inverse submonoids.
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