Structural results for semigroup subsets defined by factorization properties dependent on $$\Omega $$Ω functions

We study combinatorial properties of a class of subsets of semigroups with divisor theory motivated by the study of oscillations of counting functions of sets of algebraic integers with prescribed factorization properties.


Introduction
Semigroup subsets defined by factorization-related properties have been a subject of both structural (combinatorial) and quantitative (analytical) investigations. When the semigroup has an appropriate analytic structure (a positive-integer-valued norm with suitable properties) the counting function A(x) of its subset A may be defined as the number of non-associated elements of A with norm not exceeding x. Quantitative theory of factorizations deals with the study of the counting functions of such subsets: the size of the main term, the size of the error term, and oscillatory behaviour of the error term. The first class of problems was essentially solved in the works of Fogels [2], Narkiewicz [10-12,14,15, etc.],Śliwa [23, etc.], and Geroldinger and Halter-Koch (e.g., [4,Chap. 9]). Kaczorowski [7] obtained refined asymptotics with an estimate of the size of the error term for a number of semigroup subsets, including the set G k of elements with at most k distinct factorization lengths in the ring of integers in an algebraic number field. General treatment of the problem, in an abstract setting, is again due to Geroldinger and Halter-Koch [4,Chap. 9]. Kaczorowski and Pintz [9] and Kaczorowski and Perelli [8] studied the oscillatory behaviour of the error terms for a number of subsets in rings of algebraic integers. Their methods are directly applicable to what we call " sets of rank 0" in the sequel. Kaczorowski and Pintz [9] noted that it is possible to extend their results to G k . This was done by the author [16][17][18] and by Schmid and the author [21] except for the case of G 1 , which was only treated conditionally, by assuming some conjectures of analytical or combinatorial character. This paper is a part of a series of papers where the present author settles the case of the set G 1 in algebraic integers and, more generally, in a generalized Hilbert semigroup, as defined by Halter-Koch [5,Beispiel 4].
Let H be a semigroup with divisor theory (cf. [3,5] and [4,Definition 2.4.1]) and a finite class group G. For k ∈ N let G k denote the set of elements of H with at most k distinct factorization lengths in H . In this paper we show structural properties of G k and of a class of other semigroup subsets that allow us to show, in the sequel to this paper, using analytic results from [16,20], a theorem that implies the following (we omit the definitions of the "error term" and "oscillations" here): Theorem If H is a generalized Hilbert semigroup with more than 2 divisor classes, then for every k ∈ N the error term of the counting function of the set G k has oscillations of logarithmic frequency and size We restrict this study to subsets A of H (or of the corresponding divisor semigroup) such that the value of the characteristic function of A on an element a depends only on the values of the functions g (the number of prime divisors in a given class) on a. We call such subsets " sets". They correspond directly to "block-dependent factorization properties" considered by Geroldinger and Halter-Koch [4,Sect. 9.4]. A sufficiently regular set A may be expressed as a combination of simpler components, corresponding to what we call "cubes" in the sequel. When H is equipped with a suitable analytic structure, the complex zeta function associated to a cube is essentially a combination of products of complex powers of various L-functions multiplied by polynomials in the logarithms of these L-functions, with coefficients being complex functions regular in a larger region. This allows for the determination of the size of the main term of the counting function corresponding to the cube, and to the result that the size main term of A(x) is of the order where the constants η and δ depend on the combinatorial structure of A, specifically on what we call the rank (rk A) and degree (deg A) of A. This method was initiated by Narkiewicz [13,Theorem 9.4], cf. also Geroldinger and Halter-Koch [4,Theorem 9.4.3]. The values of the rank and degree of subsets defined by specific arithmetical properties are related to a number of interesting open problems in finite abelian groups, specifically in Zero-sum theory. For example, in the case when each divisor class contains a prime divisor, the degree of the set of irreducibles in H is the Davenport constant of the class group G [4, Chap. 6], rk G 1 equals the constant denoted as μ(G), cf., e.g., [4,Sect. 6.7], and the degrees deg G k were considered byŚliwa [24] (in an equivalent form) and Radziejewski and Schmid [21].
Building upon the methods of Kaczorowski and Pintz [9] and Kaczorowski and Perelli [8] one can also show the existence of oscillations of the error term of A(x) by showing that the associated zeta function is not regular in the half-plane s ≥ 1 2 apart from the real line. The problem is that in most cases neither the combinatorial structure of A (the expression of A as a combination of cubes), nor the analytic shape of the zeta functions of cubes, nor the multiplicities of zeros of the related L-functions are known completely. Roughly speaking, the rank of a cube affects the exponents of the complex powers of L-functions in the associated zeta function, and its degree is the degree of the polynomial in the logarithms. Our approach is to concentrate on the principal summands of zeta functions of cubes of some rank r and degree d > 0 (so that logarithms are necessarily involved) and show that similar terms do not cancel out. This motivates the notion of (r, d)-singular sets introduced in Sect. 3.4-with suitable analytical structure we are able to show the existence of oscillations precisely for such sets. The main result of the paper is the complete characterization of (r, d)-singular sets closed upon in-class divisors (Theorem 11), that allows us to treat G 1 . Essentially, previous methods allowed us to show the existence of oscillations when deg Sections 3.1 through 3.3 provide the necessary basis for the rest of the paper. A number of results and methods there are known, but we include some of the results for completeness and some, because they were not stated explicitly in the form that we need further. It seems that the rank and degree did not have a general name previously, and that they were not considered for sets other than those expressible as finite combinations of cubes. The implication

Notation
The order of a group element g is denoted as ord g. For sets A and B we denote by A B is the set of functions from a set B to a set A, and by |A| the cardinality of A. For a set X we let F(X ) denote the free (multiplicative) abelian monoid generated by a set X . For S ∈ F(X ) we put: Let G be a finite abelian group (written additively), G 0 ⊆ G. For S ∈ F(G 0 ) we put, following [4], and, for U ⊆ G 0 , S ∈ F(G 0 \U ) ⊆ F(G 0 ), y ∈ G and d ∈ N 0 , In particular we have We let S(G 0 ) denote the set of all triples (U, S, y) such that U ⊆ G 0 , S ∈ F(G 0 \U ) ⊆ F(G 0 ), y ∈ G, and y (U, S) = ∅. We refer to such non-empty sets y (U, S) as cubes (over G 0 ), and a cube contained in another cube a sub-cube of the latter. For a set U ⊆ G 0 the cube 0 (U, 1) is also denoted as B(U ). It is a submonoid of F(U ) and it is called the block monoid over U . For any subset A ⊆ F(G 0 ) we define its in-class divisor closure as For a set X we consider the set F(X ) = (N 0 ∪ {+∞}) X with the product order ≤.
We treat F(X ) as a subset of F(X ). For elements of F(X ) the order ≤ coincides with divisibility, but we still use the symbol ≤ wherever the elements of F(X ) might be involved. However, for compatibility with the notation employed for F(X ), we denote the initial element of F(X ) as 1 and, for a ∈ F(X ), We recall that a subset of a partially ordered set is called an antichain if no two of its elements are comparable. If a, b belong to a partially ordered set A, then we write For partially ordered sets A 1 and A 2 we define the lexicographic order in The product order in a product of partially ordered sets i∈I A i is of course

Basic properties of cubes
then we have This can be rewritten as the conjunction of (2) and The existence of such S implies (2). Moreover, as (4) is equivalent to S ∈ y (U 1 ∩ U 2 , lcm(S 1 , S 2 )), the condition (2) implies the equality (3).
Moreover, for every S 0 ∈ F(G 0 ) such that S 0 | S, and every S ∈ y (U, Proof 1. Let S ∈ y (U, S). Then for every d we have and the "moreover" part is by definition.

To see
. Suppose, as we may, that U 1 is not a proper subset of any of the U i , i = 2, . . . , n.
If we had Hence U i = U 1 by maximality of U 1 , and so S i = S 1 , contrary to the assumption of the triples being distinct. Hence we have y 1 (U 1 , S 1 ) The characteristic function of y 1 (U 1 , S 1 ) is thus linearly independent of the others and the assertion follows by induction.

Rank and degree
If A is a non-empty subset of F(G 0 ), we define its rank rk A as the smallest r ∈ {0, . . . , |G 0 |} such that there exist n ∈ N and (U 1 , and max i |U i | ≤ r . Such an r always exists by (1). We put rk ∅ = −∞.

Lemma 3 1. For
Proof 1. and 2. follow from the definition of rank. 3. Let r = |U |. The inequality rk y (U, S) ≤ r is trivial by the definition of rank.
The converse inequality follows from the definition and Lemma 2.3. 4. Follows from 1. and Next we define the degree deg A of a set A ⊆ F(G 0 ) as the supremum of all values of |S| over the elements (U, S, y) of the set We note that y (G 0 , 1) satisfies the condition in (5) for some y ∈ G by (1) and Lemma 3.4, so deg A ≥ 0. In particular we have deg ∅ = +∞.

For
is monotone with respect to the partial order defined by inclusion of subsets of F(G 0 ), and lexicographic order for (rk, deg) pairs.
by the definition of degree. We also have rk y i (U i , S i ) ≤ r by assumption and Lemma 3.3, and rk y i (U i , S i ) ≥ r by Lemma 3.2. Hence 5. Follows from 4., because a finite union n i=1 y i (U i , S i ) satisfying the assumptions of 4. must exist by the definition of rank.

Sets with a finite number of layers
For every A ⊆ F(G 0 ) we define l(A), the number of layers of A, as the maximum length l of an interleaved chain of divisors that a 1 , a 2 , . . . , a l ∈ A, b 1 , b 2 , . . . , b l−1 / ∈ A, and there exists some y ∈ G such that σ (a i ) = σ (b j ) = y for i = 1, . . . , l, j = 1, . . . , l − 1. In particular we have l(∅) = 0. We need one more auxiliary result before giving a complete characterization of sets with a finite number of layers.

Lemma 5 If X is a finite set, then every antichain in F(X ) is finite.
Proof For an antichain A ⊆ F(X ) let I (A) denote the set of such x ∈ X that there exists an a ∈ A that satisfies v x (a) = +∞. Suppose there exists an infinite antichain A ⊆ F(X ) and that |X | is the smallest possible number for which this happens. Suppose further that |I (A)| is the smallest possible given this X . Let a ∈ A. We have is an antichain and |X \ {x}| < |X |, so A x, j is finite by assumption, and so is A x, j , as equinumerous to A x, j . Hence A is finite, a contradiction.

iii) The characteristic function of A can be represented in the form
where n ∈ N 0 , sets are tacitly identified with their characteristic functions, the triples (U j , S j , g j ) ∈ S(G 0 ) are pairwise distinct, and γ j ∈ Z\ {0}, j = 1, . . . , n.
The representation (7) is then unique up to order. If A = ∅, then we have and γ j = 1 for all j ∈ {1, . . . , n} such that the cube g j (U j , S j ) is inclusion-maximal among others.
Proof (i) ⇒ (ii). First we reduce the general case to a specific one. The intersection of A with each subset y (G 0 , 1) in (1) may be treated independently, so we can assume A ⊆ y (G 0 , 1) for some fixed y ∈ G. We set Y = y (G 0 , 1). Let A 1 denote the set of all a 1 ∈ A such that for some a 2 , . . . , a l ∈ A and b 1 , b 2 We have l(A 1 ) = 1 and l(A\A 1 ) = l − 1, so it is enough to prove the assertion in the case l(A) = 1 and use induction. Finally, given l(A) = 1, we note that the set A = Div(A)\A satisfies Div(A ) = A , so A = Div(A)\ Div(A ). As the family of sets expressible in the form required in (ii) is closed upon set difference, it is enough to prove the assertion in the case when we need to show that for every a ∈ A there exists some b ∈ B max such that a ≤ b , so we assume otherwise. We have A ⊆ B, so, starting with b 1 = a, we can construct an increasing sequence in B: We let U denote the set of elements g ∈ G 0 such that v g (b 2 ) = +∞. We can assume that U is the largest possible. For every a ∈ Y , a ≤ sup n (b n ), and every g ∈ G 0 , we can find some n(g) ∈ N such that v g (a ) ≤ v g (b n(g) ), so a ≤ b n , where n = max g∈G 0 n(g), and a ∈ A. Therefore sup n (b n ) ∈ B. By the maximality of U we have v g sup n (b n ) < +∞ for all g ∈ G 0 \U , contradicting (10). We have shown (9) and the assertion follows, as B max is finite by Lemma 5.
(ii) ⇒ (iii), (8) etc. We assume the B i s non-empty and we can have v g (α i,1 ) = v g (α i,2 ) for each i ∈ {1, . . . , m} and g ∈ G 0 such that v g (α i,2 ) < +∞, by splitting each B i in (ii) to finitely many summands if necessary. For each B i we put for some set R i equal to a finite union of proper sub-cubes of y i (V i , β i ). We can group the B i s with equal triples (V i , β i , y i ), and possibly re-order them, so that for some m ≤ m we have . . , m , i = j, and each set R i is equal to a finite union of sub-cubes of y i (V i , β i ) of lower rank. By the inclusionexclusion principle we obtain and further where R is a combination of characteristic functions of proper sub-cubes of the We obtain (iii). Uniqueness of (7) follows from Lemma 2.4. By Lemmas 3 and 4 we have rk B i = |V i | and deg B i = |β i |, i = 1, . . . , m , so (8) follows. Finally we note that every summand of R i in (11) is a proper subset one of the cubes y i (V i , β i ), and so is every intersection i∈I B i in (11). Hence the cubes y i (V i , β i ) are the only inclusion-maximal sets in (12), and they appear with the coefficient 1 there.
(iii) ⇒ (i). Suppose l ≥ n + 1 and a 1 , a 3 , . . . , a 2l−1 ∈ A, a 2 , a 4 , . . . , a 2l−2 ∈ F(G 0 )\A are such that a 1 | a 2 | a 3 | . . . | a 2l−1 and for some y ∈ G we have σ (a i ) = y, i = 1, . . . , 2l − 1. For i = 1, . . . , 2l − 1 and j = 1, . . . , n, let c i j denote the value of the characteristic function of g j (U j , S j ) on a i , and let c 2l, j = 0. No two adjacent rows of the matrix c i j i=1,...,2l, j=1,...,n are equal as, by (7), it would place two subsequent a i s both in A or outside of A, or it would imply a 2l−1 / ∈ A. By the Dirichlet's box principle there exists some j ∈ {1, . . . , n} such that c i j = c i+1, j for at least three different values of i. However this is impossible by l( g j (U j , S j )) = 1. Hence l(A) ≤ n. Given (7) and r ∈ N 0 we call the number the rank-r degree of A. The rank-0 degree will also be called the absolute degree of

Singular sets
If A ⊆ F(G 0 ), l(A) < +∞, r ∈ N 0 and d ∈ N, then we say that A is (r, d)-singular if the unique representation (7) satisfies the following conditions: (i) There is at least one j ∈ {1, . . . , n} such that (ii) The sign of γ j is the same for every j satisfying (13).
We note that the condition (ii) is non-trivial: although the (r, d) pair is maximal among the pairs (rk g j (U j , S j ), deg g j (U j , S j )) in the product order by (i) and (iii) alone, it need not be maximal in the lexicographic order, hence the summands in (7) satisfying (13) need not be inclusion-maximal. The interplay between these two orders is really the root of the problem of showing that a set is (r, d)-singular for some suitable r and d. In the remainder of this subsection we solve this problem in the cases deg A > 0 (Fact 7) and A = Div(A) (Theorem 11).

Fact 7 If
Proof We can take r = rk A and d = deg A. Conditions (i) and (iii) hold by Proposition 6. By (8) and Lemma 2.2 any set g j (U j , S j ) in (7) satisfying U j = r is inclusion-maximal, so γ j = 1, hence condition (ii) holds as well. We

Lemma 8 A set A ⊆ F(G 0 ) is of absolute degree 0 if and only if it is a union of equivalence classes of the relation ∼.
Proof For g ∈ G, U ⊆ G 0 the set is either empty or is an equivalence class of ∼. By substituting in (7) we get the implication (⇒). To get the converse, we first note that for S ∈ F(G 0 ), g = σ (S) and U = Supp(S) we have so by the inclusion-exclusion principle and Lemma 1.1 we have If A is a union of equivalence classes of ∼, then it is a finite union, as there are only finitely many such classes in F(G 0 ). Therefore for some m ∈ N 0 and S 1 , . . . , S m ∈ F(G 0 ), where addition is in terms of characteristic functions, hence the unique representations may likewise be added, and deg 0 (A) = 0 follows from (14).

Theorem 11 Let
A be an set such that Div(A) = A. Then the following conditions are equivalent: where addition is in terms of characteristic functions, hence the unique representation (7) of A may be obtained by adding the unique representations of Ground(A) and Elev(A). We have deg 0 (Ground(A)) = 0 by Corollary 9, so A is (r, d)-singular as well.

sets in semigroups with divisor theory
Let H be a commutative, cancellative semigroup with a unit, with divisor theory ϕ : H → F(P) and a finite class group G. For a ∈ F(P) the divisor class of a is denoted as [a]. We use multiplicative notation for H . Let h = |G| and let G 0 ⊆ G be the set of classes that contain at least one prime divisor. As usual, g (a) denotes (for a ∈ F(P), g ∈ G 0 ) the number of prime divisors of a in the class g counted according to their multiplicities, cf. [4, Example 9.2.7] and, for the origins of this notation, [1,6].

A set A ⊆ F(P) is an set if and only if
In that case we define the rank, degree and the number of layers of A as rk A, deg A and l(A) respectively, and we say that A is (r, d)-singular if A is. We extend this notation and terminology also to the set ϕ −1 (A) in case A ⊆ ϕ(H ).

Elements with at most a given number of factorization lengths
We are going to show that the set G k (for a positive integer k) is (r, d)-singular for some suitable r and d unless G k = G 1 = H . First we recall some well-known facts related to G k . The set U is called half-factorial if the monoid B(U ) = 0 (U, 1) is half-factorial, i.e. every element of B(U ) has a unique length of factorization into irreducibles. The homomorphism is a transfer homomorphism [4, Definition 3.2.1], so it establishes a one-to-one correspondence between lengths of factorizations of a in H and β(a) in B(G 0 ), hence G k = β −1 (β(G k )) is an set. We have Div(G k ) = G k , in particular l(G k ) = 1. The following equivalence has appeared in several works at various levels of generality. We provide a proof for the convenience of the reader.
(iv) ⇒ (i). Every element of B(U ) is of the form β(a) for some a ∈ H , Supp(β(a)) ⊆ U . If a = a 1 . . . a n is a factorization of such a to irreducibles, then n = n i=1 k(a i ) = k(a), so a and β(a) have unique factorization length.

Theorem 13
Let k be a positive integer. The set G k is (r, d)-singular for some r ∈ N 0 and d ∈ N if and only if the set G 0 is not half-factorial.
Proof If G 0 is half-factorial, then by Lemma 12 we have β(H ) = 0 (G 0 , 1) ⊆ G 1 , so G k = G 1 = H . Suppose G 0 is not half-factorial. Then we can find an irreducible a ∈ H such that k(a) = 1. Of course a ∈ G 1 ⊆ G k . If Proof The set F min is an set, as F is. We have FF(P) = F min F(P), because divisibility is a well-order on F(P). The fact that F min is an antichain implies that l(F min ) = 1 and, using Lemma 5, alsõ where S(A) consists of all triples (U, S, y) ∈ S(G 0 ) such that for every j there is at least one g ∈ V j \U with v g (S) < v g (S j ). In that case U does not contain any of the V j and, conversely, for every U that does not contain any of the V j we have Hence deg 0 (A) = 0. Finally, if m > 1, let us fix j and g ∈ G 0 with v g (S j ) = m. Let S = g −1 S j ∈ F(G 0 ), so Supp(S) = V j , and let y = σ (S). We have S ∈ A, because S j ∈β(F min ). If [S] ∼ = y (V j , 1, 1) ⊆ A, then we would have V j ⊆ U , a contradiction. Hence deg 0 (A) > 0 and the assertion follows by Theorem 11.