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Abstract
Let H be a transfer Krull monoid over a subset G0 of an abelian group G with finite exponent. Then every non-unit can be written as a finite product of atoms, say
The set
of all possible factorization lengths k is called the set of lengths of a, and H is said to be half-factorial if
for all
We show that, if
is a non-unit and
then the smallest divisor-closed submonoid of H containing a is half-factorial. In addition, we prove that, if G0 is finite and
then H is half-factorial.
1. Introduction
Let H be a monoid. If an element has a factorization
where
and
are atoms of H, then k is called a factorization length of a, and the set
of all possible k is referred to as the set of lengths of a. The monoid H is said to be half-factorial if
for every
The study of half-factoriality was pioneered by Leonard Carlitz in the setting of algebraic number theory: he proved in [Citation4] that the ring of integers of a number field K is half-factorial if and only if the cardinality of its class group is either 1 or 2. After this, the concept of half-factoriality seemed to remain dormant for more than a decade until papers by Abraham Zaks [Citation32], Ladislav Skula [Citation28], and Jan Śliwa [Citation29] simultaneously appeared in 1976. In such papers, half-factoriality was studied in the context of Krull domains and c-monoids. Since then half-factoriality has been investigated in different classes of monoids (see [Citation3, Citation6, Citation17, Citation18]) and integral domains (see [Citation5, Citation7, Citation13, Citation19, Citation20, Citation23, Citation33]).
Given let
be the smallest divisor-closed submonoid of H containing a. Then
is half-factorial if and only if
for all
and H is half-factorial if and only if
is half-factorial for every
It is thus natural to ask:
Does there exist an integer such that, if
and
then
is half-factorial? (Note that, if
is half-factorial for some
then of course
for every
)
We answer this question affirmatively for transfer Krull monoids over finite abelian groups, and we study the smallest N having the above property (Theorems 1.1 and 1.2).
Transfer Krull monoids and transfer Krull domains are a recently introduced class of monoids and domains including, among others, all commutative Krull domains and wide classes of non-commutative Dedekind domains (see Section 2 and, for a survey, see [Citation10]).
Let H be a transfer Krull monoid over a subset G0 of an abelian group G. Then H is half-factorial if and only if the monoid of zero-sum sequences over G0 is half-factorial (in this case, we also say that the set G0 is half-factorial). It is a standing conjecture that every abelian group has a half-factorial generating set, which implies that every abelian group can be realized as the class group of a half-factorial Dedekind domain [Citation11].
Suppose now that H is a commutative Krull monoid with class group G and that every class contains a prime divisor. It is a classic result that H is half-factorial if and only if and it turns out that, also for
half-factorial subsets (and minimal non-half-factorial subsets) of the class group G play a crucial role in a variety of arithmetical questions (see [Citation12, Chapter 6.Citation7], [Citation15]). However, we are far away from a good understanding of half-factorial sets in finite abelian groups (see [Citation25] for a survey, and [Citation21, Citation22, Citation26]). To mention one open question, the maximal size of half-factorial subsets is unknown even for finite cyclic groups [Citation22]. Our results open the door to a computational approach to the study of half-factorial sets.
More in detail, denote by the infimum of all
with the following property:
If and
then
is half-factorial.
(Here, as usual, we assume ) We call
the half-factoriality index of H. If H is not half-factorial, then
is the infimum of all
with the property that
for every
such that
is not half-factorial. In particular, if G is an abelian group with
then
is the infimum of all
with the property that
For every sequence S over G, if then
for every
Theorem 1.1.
Let H be a transfer Krull monoid over a finite subset G0 of an abelian group G with finite exponent. The following are equivalent.
H is half-factorial.
G0 is half-factorial.
We observe that in general if H is half-factorial, then But if H is a transfer Krull monoid over a subset of a torsion free group, then
does not imply that H is half-factorial (see Example 2.4.1). Furthermore, for every
there exists a Krull monoid H with finite class group such that
(see Example 2.4.2).
Theorem 1.2.
Let H be a transfer Krull monoid over an abelian group G.
if and only if
If
and
, then
If G is cyclic or
, then
We postpone the proofs of Theorems 1.1 and 1.2 to Section 3.
2. Preliminaries
Our notation and terminology are consistent with [Citation12]. Let be the set of positive integers, let
and let
be the set of rational numbers. For integers
we denote by
the discrete, finite interval between a and b.
Atomic monoids. By a monoid, we mean a semigroup with identity, and if not stated otherwise we use multiplicative notation. Let H be a monoid with identity The set of invertible elements of H will be denoted by
and we say that H is reduced if
The monoid H is said to be unit-cancellative if for any two elements
each of the equations au = a or ua = a implies that
Clearly, every cancellative monoid is unit-cancellative.
Suppose that H is unit-cancellative. An element is said to be irreducible (or an atom) if
and for any two elements
u = ab implies that
or
Let
denote the set of atoms of H. We say that H is atomic if every non-unit is a finite product of atoms. If H satisfies the ascending chain condition on principal left ideals and on principal right ideals, then H is atomic [Citation9, Theorem 2.6]. If
and
where
and
then k is a factorization length of a, and
denotes the set of lengths of a. It is convenient to set
for all
Let H and B be atomic monoids. The homomorphism is called a weak transfer homomorphism if it satisfies the following two properties.
(T1) and
(WT2) If and
then there exist
and a permutation
such that
and
for each
A transfer Krull monoid is a monoid H having a weak transfer homomorphism where
is the monoid of zero-sum sequences over a subset G0 of an abelian group G. If H is a commutative Krull monoid with class group G and
is the set of classes containing prime divisors, then there is a weak transfer homomorphism
Beyond that, there are wide classes of non-commutative Dedekind domains having a weak transfer homomorphism to a monoid of zero-sum sequences ([Citation31, Theorem 1.1], [Citation30, Theorem 4.4]). We refer to [Citation10, Citation16] for surveys on transfer Krull monoids. If
is a weak transfer homomorphism, then sets of lengths in H and in
coincide [Citation2, Lemma 2.7] and thus the statements of Theorems 1.1 and 1.2 can be proved in the setting of monoids of zero-sum sequences.
Monoids of zero-sum sequences. Let G be an abelian group and let be a non-empty subset. Then
denotes the subgroup generated by G0. In additive combinatorics, a sequence (over G0) means a finite unordered sequence of terms from G0 where repetition is allowed, and (as usual) we consider sequences as elements of the free abelian monoid with basis G0. Let
be a sequence over G0. We call
The sequence S is said to be
zero-sum free if
a zero-sum sequence if
a minimal zero-sum sequence if it is a nontrivial zero-sum sequence and every proper subsequence is zero-sum free.
The set of zero-sum sequences is a submonoid, and the set of minimal zero-sum sequences is the set of atoms of
For any arithmetical invariant
defined for a monoid H, we write
instead of
In particular,
is the set of atoms of
and
Let G be an abelian group. We denote by the exponent of G which is the least common multiple of the orders of all elements of G. If there is no least common multiple, the exponent is taken to be infinity. Let
and let
be an r-tuple of elements of G. Then
is said to be independent if
for all
and if for all
an equation
implies that
for all
Suppose G is finite. The r-tuple
is said to be a basis of G if it is independent and
For every
we denote by Cn an additive cyclic group of order n. Since
is the rank of G and
is the exponent of G.
Let be a non-empty subset. For a sequence
we call
For the relevance of cross numbers in the theory of non-unique factorizations, see [Citation22, Citation24, Citation27] and [Citation12, Chapter 6].
The set G0 is called
half-factorial if the monoid
is half-factorial;
non-half-factorial if the monoid
is not half-factorial;
minimal non-half-factorial if G0 is not half-factorial but all its proper subsets are;
an LCN-set if
for all atoms
The following simple result [Citation12, Proposition 6.7.3] will be used throughout the article without further mention.
Lemma 2.1.
Let G be a finite abelian group and a subset. Then the following statements are equivalent.
G0 is half-factorial.
for every
for every
Lemma 2.2.
Let G be a finite group, let be a subset, let S be a zero-sum sequence over G0, and let A be a minimal zero-sum sequence over G0.
If
, then
If there exists a zero-sum subsequence T of S such that
, then
If
and
is minimal over all minimal zero-sum sequences over G0, then
Proof.
1. Suppose and let
where
and
Then
which implies that
It follows by
that
2. Suppose T is a zero-sum subsequence of S with It follows by
that
3. Suppose and
is minimal over all minimal zero-sum sequences over G0. Let
Then there exist
and minimal zero-sum sequences
such that
Since
we have
and hence
□
For commutative and finitely generated monoids, we have the following result.
Proposition 2.3.
Let H be a commutative unit-cancellative monoid. If is finitely generated, then
is finite.
Proof.
We may assume that H is reduced and not half-factorial. Suppose H is finitely generated and suppose where
Set
Then A0 is finite and hence there exists
such that
for all
with
not half-factorial. Let
such that
is not half-factorial. It suffices to show that
Suppose
where
Set
and
Then a0 divides a and
is not half-factorial, whence
and
□
If G0 is a finite subset of an abelian group, then is finitely generated [Citation12, Theorem 3.4.2] and thus
We refer to [Citation8, Sections 3.2 and 3.3] and [Citation14] for semigroups of ideals and semigroups of modules that are finitely generated unit-cancellative but not necessarily cancellative.
Example 2.4.
The following examples will help us to illustrate some important points.
Let (e1, e2) be a basis of
and let
Then
Since
we obtain G0 is not half-factorial. Furthermore, we have G1 is half-factorial for every non-empty proper subset
Let
If
then
and
is not half-factorial. If
then
is half-factorial and
Therefore
Let G be a cyclic group with order n and let
with
where
Set
Then G0 is not half-factorial. Let
Then
is not half-factorial and
whence
Let
such that
is not half-factorial. Then
and A0 divides A, whence
Therefore
Let
and let (e1, e2) be a basis of G. Set
Then G1 is not half-factorial. Let
Then
is not half-factorial and
whence
Let
such that
is not half-factorial. Then
and A1 divides A, whence
Therefore
Let H be a bifurcus monoid (i.e.,
for all
). For examples, see [Citation1, Examples 2.1 and 2.2]. Since for every
we have
it follows that
and
is the minimal integer
such that
for all
Therefore
if and only if there exists
such that
Let
be a non-half-factorial finitely primary monoid of rank s and exponent α (see [Citation12, Definition 2.9.1]). For every
we define
where
and
Let
Since H is primary, we have
is not half-factorial. Thus
is the minimal integer
such that
for all
Suppose
with
Then
and
whence
If and
then H is bifurcus and hence
Suppose s = 1 and
Let
Then
divides b4. It follows by
that
whence
If
then
divides b3 and hence
If
then b is an atom and
whence
If
then
Put all together, if
then
Otherwise
3. Proof of main theorem
Proposition 3.1.
Let be a non-half-factorial subset and let S be a zero-sum sequence over G0 with
If G0 is an LCN-set, then
If
, then
If G0 is a minimal non-half-factorial subset, then
If
, then
Proof.
1. Suppose G0 is an LCN-set. Since G0 is not half-factorial, there exists a minimal zero-sum sequence T over G0 such that Note that T is a subsequence of
Then there exist
such that
Thus
The assertion follows by
2. Suppose and let
If G0 is an LCN-set, the assertion follows by 1. Suppose there exists a minimal zero-sum sequence T over G0 with
Let
be the minimal zero-sum sequence over G0 such that
is minimal. If
say
then
Thus
a contradiction to the minimality of
Therefore
and hence
It follows that
3. Suppose that G0 is a minimal non-half-factorial set. If S has a minimal zero-sum subsequence A with then the assertion follows by Lemma 2.2. If G0 is an LCN-set, then the assertion follows from 1 and Lemma 2.2.2. Therefore we can suppose
and suppose there exists a minimal zero-sum sequence T over G0 with
Let be the minimal zero-sum sequence over G0 such that
is minimal. The minimality of G0 implies that
for all
After renumbering if necessary, we let
By Lemma 2.2.3,
If
divides
then the assertion follows by Lemma 2.2.2. Suppose
Let
Thus for each
we have
which implies that
and
Let such that
Therefore for every
we have
Note that for every
we have
It follows by
that
By the minimality of G0, we have
is half-factorial which implies that
Therefore
is an integer, a contradiction to
4. Let Suppose
is not half-factorial. If
is an LCN-set, then the assertion follows by Proposition 3.1.1 and Lemma 2.2.2. Otherwise there exists a minimal zero-sum sequence A over
such that
We may assume that
is minimal over all minimal zero-sum sequences over
and that
for some
Thus by Lemma 2.2.3, we have
The definition of G1 implies that
and hence the assertion follows.
Suppose is half-factorial. Then G1 is non-empty and hence
for some
If G0 is an LCN-set, then the assertion follows by Proposition 3.1.1 and Lemma 2.2.2. Otherwise there exists a minimal zero-sum sequence A over G0 such that
We may assume that
is minimal over all minimal zero-sum sequences over G0 and that
for some
Thus by Lemma 2.2.3, we have
For every
we obtain
If
then
and hence
Otherwise for every we have
Therefore
for all
which implies that A divides
Thus there exists a zero-sum sequence W over
such that
Since
is half-factorial, we obtain
is an integer, a contradiction to
□
Proof of Theorem 1.1.
By the definition of transfer Krull monoid, it suffices to prove the assertions for and hence H is half-factorial if and only if G0 is half-factorial. If G0 is half-factorial, it is easy to see that
and
Therefore we only need to show that (b) implies (c) and that (d) implies (c).
(b) (c) Suppose
and assume to the contrary that G0 is not half-factorial. Then there exists
such that
whence
is not half-factorial. Therefore
a contradiction.
(d) (c) Suppose
and assume to the contrary that G0 is not half-factorial. If G0 is an LCN set, then Proposition 3.1.1 implies that
a contradiction. Thus there exists an atom
with
and we may assume that
is minimal over all atoms of
Let
Then by Lemma 2.2.3, we have
a contradiction to
□
Proof of Theorem 1.2.
By the definition of transfer Krull monoid, it suffices to prove all assertions for
1. Suppose If
then 2 implies that
If
then
is half-factorial and hence
Suppose If there exists an element
with
then
is an atom for every
Since
is not half-factorial and
for every
we obtain that
for every
that is,
Otherwise G is torsion. Then there exists a sequence
with
and
It follows by 1 that
for all
that is,
2. If G is an elementary 2-group and e1, e2 are two independent elements, then is not a half-factorial set and
which implies that
Otherwise there exists an element
with
Since
is not half-factorial and
we obtain
Let S be a zero-sum sequence over G such that is not half-factorial. In order to prove
we show that
Set
If G0 is an LCN-set, the assertion follows by Proposition 3.1.1. Suppose there exists an atom
with
Let
be such that
is minimal over all minimal zero-sum sequences over G0 and set
where
and
are pairwise distinct elements. If there exists
such that
then
divides
and hence
for some non-empty sequence
Thus
a contradiction to the minimality of
Therefore
After renumbering if necessary, we assume
Then
which implies that
divides
The assertion follows by Lemma 2.2.3.
3(a). Suppose that G is cyclic and that with
We will show that
Let S be a zero-sum sequence over G such that is not half-factorial. It suffices to show that
If
then the assertion follows from Proposition 3.1.4. Suppose
Then there exist distinct
such that
We may assume that g1 = kg2 for some
with
It follows by
that
is not half-factorial. By Proposition 3.1.2, we obtain that
3(b). Suppose G is a finite abelian group with We need to prove that
Let S be a zero-sum sequence over G such that
is not half-factorial. It suffices to show that
If is an LCN-set, the assertion follows by Proposition 3.1.1. Thus there is a minimal zero-sum sequence W over
such that
By Proposition 3.1.2 and Lemma 2.2.2, we have
Suppose
Since
it follows by Lemma 2.2 that
and hence
Therefore we may assume that
whence
It follows that
We distinguish two cases according to
Case 1.
Then, and for all
with
we have that
Let W0 be an atom over with
Then
where
are pairwise distinct and
with
We may assume T is zero-sum free. Otherwise where T0 is a zero-sum sequence and
is zero-sum free. We can replace S by
since
implies that
Therefore S is a product of at most three atoms and every term of S has order 5.
Assume to the contrary that that is,
Since
is a zero-sum subsequence of S5, we obtain that
is an atom. Note that
Suppose S is an atom. Then and hence
that is,
It follows by
that
a contradiction.
Suppose S is a product of two atoms. Then and hence
that is,
It follows by
that
a contradiction.
Suppose S is a product of three atoms. Then which implies that
such that
is zero-sum for all
Since
we obtain
and hence
Therefore
that is,
It follows by
that
a contradiction.
Case 2.
Let W be an atom over with
If
then
and hence W must be of the form
where
are pairwise distinct. Since
we obtain that
It follows from the fact that
divides
that
If then
or 4 and hence W can be written in one of the following ways.
where
are pairwise distinct.
where
are pairwise distinct.
where
are pairwise distinct.
Suppose (i) holds. Then Since
we obtain that
It follows from the fact that
divides
that
Suppose (ii) holds. Then Thus
and hence
a contradiction.
Suppose (iii) holds. Then Therefore
is zero-sum. If
is not a minimal zero-sum sequence, then
and hence
If W0 is minimal zero-sum, then
implies that
and hence
□
Acknowledgments
We thank the referee for very careful reading and for providing valuable suggestions.
Additional information
Funding
References
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