Length-Factoriality and Pure Irreducibility

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arXiv:2210.06638v2 [math.AC] 20 Mar 2024

This is an Accepted Manuscript of an article

published by Taylor & Francis in Communications in Algebra on 20 Mar 2023,

available at: https://www.tandfonline.com/doi/10.1080/00927872.2023.2187629

Length-Factoriality and Pure Irreducibility

ALAN BU1, JOSEPH VULAKH2, AND ALEX ZHAO3

Abstract. An atomic monoid Mis called length-factorial if for every non-invertible

element x∈M, no two distinct factorizations of xinto irreducibles have the same

length (i.e., number of irreducible factors, counting repetitions). The notion of length-

factoriality was introduced by J. Coykendall and W. Smith in 2011 under the term

“other-half-factoriality”: they used length-factoriality to provide a characterization of

unique factorization domains. In this paper, we study length-factoriality in the more

general context of commutative, cancellative monoids. In addition, we study factor-

ization properties related to length-factoriality, namely, the PLS property (recently

introduced by Chapman et al.) and bi-length-factoriality in the context of semirings.

1Phillips Exeter Academy, Exeter, NH 03833

2Paul Laurence Dunbar High School, Lexington, KY 40513. joseph@vulakh.us

3Lakeside School, Seattle, WA 98125

2020 Mathematics Subject Classiﬁcation. Primary: 13F15, 13A05; Secondary: 16Y60.

Key words and phrases. length-factoriality, factorization, atomicity, unique factorization, other-half-

factoriality, pure irreducible, semidomain, ﬁnite-rank monoid, monoid algebra.

2A. BU, J. VULAKH, AND A. ZHAO

1. Introduction

A (multiplicative) cancellative commutative monoid Mis called atomic if every non-

invertible element of Mfactors into atoms (i.e., irreducible elements), and an integral

domain is atomic if its multiplicative monoid is atomic. Every submonoid of a free

commutative monoid is atomic, although some elements may have multiple factorizations

into atoms. It follows from the deﬁnitions that UFDs are atomic domains. In addition,

Noetherian domains and Krull domains are well-studied classes of atomic domains that

may not be UFDs. The phenomenon of multiple factorizations in atomic monoids and

domains has received a great deal of investigation during the last three decades (see [12]

and [2] for recent surveys). In this paper, we study atomicity and the phenomenon of

multiple factorizations in the settings of monoids, integral domains, and semidomains,

putting special emphasis on the property of length-factoriality and on certain special

types of atoms, called pure atoms by Chapman et al. in [6], that appear in length-

factorial monoids.

An atomic monoid Mis called a length-factorial monoid provided that no two dis-

tinct factorizations of the same non-invertible element of Mhave the same length (i.e.,

number of irreducible factors, counting repetitions). The notion of length-factoriality

was introduced and ﬁrst studied in 2011 by Coykendall and Smith in [9] under the term

“other-half-factoriality”. In their paper, they proved that an integral domain is length-

factorial if and only if it is a UFD. By contrast, it is well known that there are abundant

classes of length-factorial monoids that do not satisfy the unique factorization property.

More recent studies of length-factoriality have been carried out by Chapman et al. in [6]

and by Geroldinger and Zhong in [11]. In Section 3, we discuss length-factoriality in the

context of monoids, proving that every length-factorial monoid Mis an FFM (i.e., M

is atomic and every non-invertible element of Mhas only ﬁnitely many divisors up to

associates). We conclude Section 3by investigating the structure of LFMs and the num-

ber of pure atoms that can appear in an atomic monoid, showing that for any prescribed

pair (m, n)∈N2, there is an atomic monoid having precisely mpurely long atoms and

npurely short atoms.

An atom a1of an atomic monoid Mis called purely long if the fact that a1···aℓ=

a′

1···a′

mfor some atoms a2,...,aℓ, a′

1,...,a′

mof Mwith aiand a′

jnot associates for any

i,j, implies that ℓ > m. A purely short atom is deﬁned similarly. It turns out that every

length-factorial monoid that does not satisfy the unique factorization property has both

purely long and purely short atoms [6, Corollary 4.6]. On the other hand, it was proved

in [6, Theorem 6.4] that an integral domain cannot contain purely long and purely short

atoms simultaneously and, in addition, the authors gave examples of Dedekind domains

having purely long (resp., purely short) atoms, but not purely short (resp., purely long)

atoms. In Section 4, we prove that monoid algebras with rational exponents have neither

purely long nor purely short atoms.

LENGTH-FACTORIALITY AND PURE IRREDUCIBILITY 3

Section 5is devoted to presenting a still unanswered question about length-factoriality

and pure irreducibility in the setting of semidomains. A semidomain is an additive

submonoid of an integral domain that is closed under multiplication (unlike subrings,

in a semidomain, some elements may not have additive inverses). It is still not known

whether N0is the only semidomain with both its additive and its multiplicative monoids

being length-factorial. To assist in the study of length-factoriality in semidomains, we

extend to the context of semidomains the result of [6] that no integral domain has both

purely long and purely short atoms.

2. Background

In this section, we introduce some terminology and deﬁnitions related to the atomicity

of cancellative commutative monoids. We use Nand N0to denote the sets of positive

and non-negative integers, respectively. We denote the set of rational and real numbers

by Qand R, respectively. For a, b ∈Z, we denote the set of integers between aand b,

inclusive, as follows:

Ja, bK:= {n∈Z|a≤n≤b}.

Over the course of this paper, we tacitly assume that the term monoid will always

mean cancellative commutative monoid. Let Mbe a monoid. Since Mis assumed to be

commutative, we will use additive notation, where “+” denotes the operation of Mand 0

denotes the identity element. We call the invertible elements of Munits, and the set of

all units of Mis denoted by U(M). The monoid Mis called reduced if U(M) = {0}.

If Mis generated by a set S, then we write M=hSi.

The quotient group or the diﬀerence group of Mis the set of diﬀerences of elements of

M(that is, the unique, up to isomorphism, abelian group gp(M) satisfying that every

abelian group containing a copy of Mmust also contain a copy of gp(M)) [13, pp. 5–6].

The monoid Mis called torsion-free if, for any n∈Nand x, y ∈M, the equality nx =ny

implies that x=y. One can easily show that Mis torsion-free if and only if gp(M) is

a torsion-free abelian group. By deﬁnition, the rank of Mis the rank of gp(M) as a

Z-module, that is, the dimension of the vector space Q⊗Zgp(M).

For x, y ∈M, the element ydivides xin M, denoted y|Mx, if there exists an element

z∈Msuch that x=y+z. We will denote such a divisibility relation by just y|xif

the monoid is clear from the context. Two elements xand yof Mare associates if each

of xand ydivides the other. A subset Sof Mis divisor-closed if Scontains all divisors

in Mof all elements of S.

The monoid Mis called a numerical monoid if Mis an additive submonoid of N0and

N0\Mis ﬁnite. Numerical monoids have been study for many decades motivated in part

by their connection with the Frobenius coin problem. A Puiseux monoid is an additive

submonoid of Q≥0. Puiseux monoids, being a crucial playground for counterexamples in

factorization theory and commutative algebra (via monoid domains), have been actively

investigated for the last few years (see [10] and references therein).

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arXiv:2210.06638v2[math.AC]20Mar2024ThisisanAcceptedManuscriptofanarticlepublishedbyTaylor&FrancisinCommunicationsinAlgebraon20Mar2023,availableat:https://www.tandfonline.com/doi/10.1080/00927872.2023.2187629Length-FactorialityandPureIrreducibilityALANBU1,JOSEPHVULAKH2,ANDALEXZHAO3Abstract.Anatomicm...

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