Survey Article: Graphical representations of factorizations in commutative rings

A study of factorizations of elements in monoids via related graphical structures

“…(2) This statement follows immediately from the hypotheses and Theorem 4.2 (2). (1) H is a factorial.…”

“…In this final section we consider bounds on the elasticity of an element obtained from the graphical structures defined in Section 2. In particular, we give some improvements to the bounds given in [2] and [4]. Recall that if x is a nonunit of a BFM H, L(x) = {t : x = a 1 · · · a t with each a i irreducible} is the set of lengths of x and that ρ(x) = max L(x) min L(x) is the elasticity of x.…”

“…We note that ρ(x) = 1 for all x ∈ H if and only if H is a HFM and thus the elasticity gives a measurement of how non-unique factorizations of x can be. Before giving our improvements, we recall the bounds on elasticity given in [2] and [4].…”

“…In fact, this idea of graphically representing factorizations was introduced earlier in a different context to compute minimal representations of numerical semigroups (see [8]). Since the seminal paper [5] where the authors were able to use graphs to characterize certain classes of domains including unique factorization domains (UFDs), finite factorization domains (FFDs), and half factorial domains (HFDs), several authors (see [1,2,3,4]) have used this idea in an attempt to understand how elements factor as a product of irreducibles. Thus far, three graphical structures have been introduced in the cancellative setting; irreducible divisor graphs [5], compressed irreducible divisor graphs [1], and irreducible divisor simplicial complexes [4].…”

“…Thus far, three graphical structures have been introduced in the cancellative setting; irreducible divisor graphs [5], compressed irreducible divisor graphs [1], and irreducible divisor simplicial complexes [4]. Our goal in the current manuscript is to (1) introduce a fourth structure, the compressed irreducible divisor simplicial complex, (2) compare and contrast the four graphical structures, (3) extend known results using the new object and how it relates to the other three, and (4) study unique and non-unique factorization in commutative cancellative monoids via these four graphical structures.…”