Factorization theory in commutative monoids

Abstract

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

Lattices over Bass Rings and Graph Agglomerations

“…We introduce those most important to this work. For additional invariants and further details we refer to the survey by Geroldinger-Zhong [32] and the references cited therein, as well as the monograph [28]. For the rest of this section we assume the following standing hypotheses.…”

“…AAMPs) are essentially (unions of) arithmetic progressions, possibly with some gaps at the beginning and/or end of the sequence. Precise definitions can be found in [32,Section 2].…”

“…We recall the main finiteness results as discussed in [32,Sections 3 and 5]; see also [28,Corollary 3.4.13] for more. In fact, many of these results hold for all finitely generated [cancellative] monoids.…”

“…Remark 4.10 A divisor theory of a Krull monoid gives rise to a transfer homomorphism to a monoid of zero-sum sequences B(G 0 ), where G 0 is a subset of the divisor class group G (see the surveys [10,26,32]). In this way, to every a ∈ A(G), we can associate a zerosum sequence θ(a) ∈ B(G 0 ).…”

“…These types of questions arose at first in the factorization theory of integral domains and monoids; we mention the recent surveys [32,33], monographs [22,28], and proceedings [2,11,12]. For monoids of modules, this perspective was pursued in [5,6,8,9,14,21] for some classes of rings; also see the survey [10].…”