Length-Factorial Property in Monoids

• Length-factorial property is defined in atomic commutative cancellative monoids as the condition that any two distinct factorizations of a nonunit have different lengths.
• This property is characterized by a unique master relation and the presence of pure irreducible atoms, which govern nonunique factorization phenomena and influence invariants like elasticity and delta sets.
• In integral domains and Krull monoids, the length-factorial condition forces unique factorization, highlighting a contrast between abstract monoid theory and classical factorization in algebraic structures.

The length-factorial property is a factorization-theoretic condition on an atomic commutative cancellative monoid $M$: for every nonunit $x$, no two distinct factorizations of $x$ into atoms have the same length. Equivalently, the length map on the factorization set $\mathsf{Z}(x)$ is injective. Introduced by J. Coykendall and W. Smith under the name “other-half-factoriality,” the property is strictly weaker than unique factorization in general monoids, but in integral domains it characterizes unique factorization. Subsequent work developed structural criteria in commutative monoids, linked the property to pure irreducibility and catenary invariants, and obtained a sharp classification in the Krull setting (Bu et al., 2022, Chapman et al., 2021, Geroldinger et al., 2021).

1. Definition and basic framework

Let $M$ be an atomic commutative cancellative monoid, written multiplicatively, with group of units $M^\times$. An element $a \in M \setminus M^\times$ is an atom if every decomposition $a = xy$ forces one of $x,y$ to be a unit. Fixing representatives of associate classes, the free commutative monoid on the atoms maps onto $M$; for $x$0, the factorization set $x$1 consists of all formal factorizations of $x$2, and for $x$3, the length $x$4 is the number of atoms occurring in $x$5. The associated set of lengths is

$x$6

The monoid $x$7 is length-factorial if

$x$8

Thus distinct factorizations of the same element are distinguished solely by length (Bu et al., 2022).

This condition is formally dual to half-factoriality. A monoid is half-factorial when every element has exactly one factorization length, that is, $x$9 for all $x$0; it is factorial, or a unique factorization monoid, when $x$1 for all $x$2. One always has

$x$3

but length-factoriality and half-factoriality are independent in general monoids. In particular, length-factoriality permits multiple factorizations of a given element, provided their lengths are pairwise distinct (Chapman et al., 2021).

Factorization relations provide a convenient language for the property. A relation $x$4 with $x$5 is balanced if $x$6 and unbalanced otherwise; it is irredundant if no atom appears on both sides. Length-factoriality excludes all nontrivial balanced relations. This makes the geometry of $x$7, the factorization congruence of the factorization homomorphism, unusually rigid and places length-factoriality between the broad nonunique-factorization regime and the degenerate factorial regime (Bu et al., 2022, Chapman et al., 2021).

2. Structural characterizations in commutative monoids

A central characterization states that for an atomic, reduced commutative cancellative monoid $x$8 that is not factorial, the following are equivalent: $x$9 is length-factorial; there exists an atom $\mathsf{Z}(x)$0 such that both $\mathsf{Z}(x)$1 and $\mathsf{Z}(x)$2 are integrally independent subsets of the Grothendieck group $\mathsf{Z}(x)$3; and the factorization congruence $\mathsf{Z}(x)$4 is nontrivial and cyclic, generated by a single factorization relation. In this sense, length-factoriality is equivalent to the entire relation theory being generated by one irredundant unbalanced relation (Chapman et al., 2021).

The relevant generator is a master factorization relation. A relation $\mathsf{Z}(x)$5 is a master relation if every irredundant unbalanced relation is of the form $\mathsf{Z}(x)$6 or $\mathsf{Z}(x)$7 for some $\mathsf{Z}(x)$8. A proper length-factorial monoid—one that is length-factorial but not factorial—admits such an unbalanced master relation, and in that case the only master relations are $\mathsf{Z}(x)$9 and its reverse. This formulation makes explicit that all nonunique factorization phenomena in a proper length-factorial monoid are controlled by a single primitive asymmetry between two factorization patterns (Chapman et al., 2021, Bu et al., 2022).

Several rigidity consequences follow. If $M$0 is generated by two elements, then $M$1 is cyclic and $M$2 is length-factorial. More generally, if $M$3 is a proper length-factorial monoid of finite rank $M$4, then $M$5, so every finite-rank length-factorial monoid is finitely generated. At the opposite extreme, hereditary length-factoriality is extremely restrictive: a monoid $M$6 has every submonoid length-factorial if and only if $M$7 is a torsion abelian group (Chapman et al., 2021, Bu et al., 2022).

These descriptions show that length-factoriality is not merely a condition on sets of lengths; it is effectively a presentation-theoretic condition on the relation module. This suggests why the property is rare in ambient categories with rich divisor theory, but relatively abundant in specially constructed monoids with a single governing relation.

3. Purely long and purely short irreducibles

The theory of pure irreducibility isolates how individual atoms behave across unbalanced relations. An atom $M$8 is purely long if whenever it occurs in one side of an irredundant unbalanced relation, that side is necessarily the longer side; purely short is defined dually. The corresponding sets are denoted $M$9 and $M^\times$0. A monoid with both sets nonempty is said to have the PLS property. In a proper length-factorial monoid, the pure atoms are precisely the atoms that occur on the master relation, and every proper length-factorial monoid is a PLS monoid (Bu et al., 2022, Chapman et al., 2021).

A key structural fact is that if $M^\times$1 is pure and $M^\times$2 is an irredundant factorization relation, then $M^\times$3 appears in one of $M^\times$4 if and only if the relation is unbalanced. Thus pure atoms detect exactly the non-half-factorial part of the factorization theory. In particular, $M^\times$5 and $M^\times$6 are finite, and in PLS monoids the “pure part” can be separated from the half-factorial remainder: if $M^\times$7 is a PLS monoid, then $M^\times$8 with $M^\times$9, where $a \in M \setminus M^\times$0 is half-factorial, $a \in M \setminus M^\times$1 is a finitely generated proper length-factorial monoid, and the atoms of $a \in M \setminus M^\times$2 are exactly the pure atoms (Chapman et al., 2021).

The existence theory is particularly strong. Given $a \in M \setminus M^\times$3, positive integers $a \in M \setminus M^\times$4 with $a \in M \setminus M^\times$5, the endpoint conditions $a \in M \setminus M^\times$6 if $a \in M \setminus M^\times$7 and $a \in M \setminus M^\times$8 if $a \in M \setminus M^\times$9, and the strict inequality $a = xy$0, there exists a length-factorial monoid with purely long atoms $a = xy$1, purely short atoms $a = xy$2, and unbalanced master relation

$a = xy$3

Consequently, for every pair $a = xy$4, there exists a monoid with $a = xy$5 and $a = xy$6 (Bu et al., 2022).

The converse fails in general: the PLS property does not imply length-factoriality. This is already visible in explicit finitely generated examples, including a monoid in $a = xy$7 with a purely long atom and a purely short atom that is not length-factorial, and in direct-product constructions that are PLS and finite-factorization but admit two distinct equal-length factorizations of the same element (Chapman et al., 2021, Bu et al., 2022).

4. Arithmetic consequences and factorization invariants

Length-factoriality has immediate finiteness consequences. If $a = xy$8 is length-factorial, then $a = xy$9 is a finite factorization monoid: every element has only finitely many factorizations, and hence only finitely many lengths. Since the length map is injective on each $x,y$0, one has

$x,y$1

for every nonunit $x,y$2. Proposition 3.1 of (Bu et al., 2022) proves this implication directly by showing that, for a fixed element $x,y$3, every atom dividing $x,y$4 must already occur in one of the two shortest factorizations of $x,y$5; otherwise equal-length factorizations can be manufactured, contradicting length-factoriality (Bu et al., 2022).

This finiteness propagates to standard arithmetical invariants. For each $x,y$6, the elasticity

$x,y$7

is finite, and the delta set $x,y$8 is finite because $x,y$9 is finite. In the cancellative nonfactorial setting, the global elasticity $M$0 is finite and accepted, while the set of distances satisfies $M$1. Length-factoriality therefore enforces a highly constrained length structure: lengths may vary, but they vary along a one-dimensional combinatorial pattern generated by a single unbalanced relation (Geroldinger et al., 2021).

The property also admits a clean interpretation via catenary theory. The equal catenary degree satisfies $M$2 if and only if $M$3 is length-factorial. If $M$4 is a proper length-factorial monoid with master relation $M$5, then

$M$6

Moreover, $M$7 has exactly one Betti element up to associates; if $M$8, then $M$9, the relation graph on $x$00 has exactly two connected components, and every other element has a single relation class. Thus the entire nonunique-factorization theory is concentrated at a unique critical element (Chapman et al., 2021).

A common misconception is that length-factoriality should force near-factorial behavior across all invariants. The formulas above show a subtler picture: nonunique factorization can persist, but it is forced into a very small and explicitly computable part of the arithmetic.

5. Domains, Krull monoids, and semiring-related settings

In integral domains, length-factoriality collapses to unique factorization. Coykendall and Smith showed that an integral domain $x$01 is length-factorial if and only if it is a UFD. The later monoid-theoretic explanation is that every proper length-factorial monoid is PLS, whereas an atomic domain cannot simultaneously contain purely long and purely short irreducibles. The same obstruction persists for semidomains: if $x$02 is an atomic semidomain, then either $x$03 or $x$04, and therefore

$x$05

Proper length-factorial monoids therefore occur naturally in abstract monoid theory, but not as multiplicative monoids of integral domains or semidomains (Chapman et al., 2021, Bu et al., 2022).

An analogous exclusion holds for monoid algebras over Puiseux monoids. If $x$06 is a field and $x$07 is an atomic Puiseux monoid, then the monoid algebra $x$08 has no pure irreducibles: $x$09 This rules out the PLS property in that class and shows that passage to monoid algebras can destroy the pure-atom structure present in the original monoid (Bu et al., 2022).

For Krull monoids, the classification is sharper and is expressed through divisor class groups and block monoids. If $x$10 is a Krull monoid, then $x$11, where $x$12 is reduced, Krull, and has no prime elements; $x$13 is length-factorial if and only if $x$14 is length-factorial. The nonfactorial length-factorial case occurs exactly when every class containing prime divisors contains precisely one such prime and $x$15 is isomorphic to a block monoid $x$16 whose atoms are of the form

$x$17

with a single defining relation

$x$18

The associated class group has the form $x$19, with $x$20 (Geroldinger et al., 2021).

This classification has strong corollaries. Krull monoids with the approximation property are length-factorial if and only if they are factorial; in particular, the same is true for Krull domains, Dedekind domains, additively regular commutative Krull rings, and normalizing Krull rings. If every nonzero class contains a prime divisor, then length-factoriality is possible only in very small class groups: specifically, only when $x$21 or $x$22, together with the corresponding block-monoid condition (Geroldinger et al., 2021).

6. Examples, low-rank classifications, and open directions

Examples separating adjacent factorization properties are central to the subject. One construction takes $x$23 and $x$24; then $x$25 is atomic, a finite factorization monoid, and a PLS monoid, but not length-factorial because the element $x$26 has two distinct factorizations of length $x$27. Another family shows that for $x$28, the numerical monoid $x$29 is a finite factorization monoid but not a PLS monoid. A further modification, replacing $x$30 by $x$31, gives a PLS monoid that is not an FFM. These examples clarify that LFM, PLS, and FFM are genuinely different conditions (Bu et al., 2022).

In rank $x$32, the theory becomes very explicit. For an atomic Puiseux monoid $x$33, the following are equivalent: $x$34 is a proper length-factorial monoid; $x$35 is a PLS monoid; $x$36 and $x$37; and $x$38. When these conditions hold, the purely long and purely short sets are singletons, namely $x$39 and $x$40. For numerical monoids, this yields the criterion that $x$41 if and only if the monoid has exactly two atoms (Chapman et al., 2021).

A concrete model of prescribed asymmetry is furnished by the relation

$x$42

which arises from the realization theorem with $x$43, $x$44, $x$45, $x$46, $x$47. The resulting monoid is length-factorial, $x$48 are purely long, $x$49 is purely short, and the element $x$50 has exactly two factorizations, of lengths $x$51 and $x$52. This example is representative of the general principle that proper length-factorial monoids can be engineered with arbitrarily prescribed numbers of pure atoms and a chosen master relation (Bu et al., 2022).

Open problems are concentrated in semiring-related settings. Baeth–Chapman–Gotti asked whether $x$53 is the only positive subsemidomain of $x$54 that is bi-length-factorial, meaning that both the additive and multiplicative monoids are length-factorial. A related conjecture states that a positive semidomain is bi-length-factorial if and only if it is bi-UFS. For the exponentiation construction $x$55 from a Puiseux monoid $x$56, it is known that if $x$57 is bi-length-factorial, then $x$58 must be two-generated of the form $x$59 with $x$60 coprime. This sharply narrows the search space for counterexamples and suggests that bi-length-factoriality may be substantially more rigid than ordinary length-factoriality (Bu et al., 2022).

The modern picture is therefore bifurcated. On one side, abstract commutative monoids admit many proper length-factorial examples, all governed by a single unbalanced master relation and a tightly controlled pure-atom structure. On the other side, in domains, semidomains, and broad Krull contexts, the property becomes so restrictive that it collapses to factoriality. This tension between abundance in free-standing monoid theory and rigidity in ambient algebraic categories is the defining feature of the subject.