A Characterization of Seminormal C-Monoids

Bollettino dell’Unione Matematica Italiana https://doi.org/10.1007/s40574-019-00194-9

A characterization of seminormal C-monoids

Alfred Geroldinger1 · Qinghai Zhong1

Received: 27 September 2018 / Accepted: 30 January 2019 © The Author(s) 2019

Abstract It is well-known that a C-monoid is completely integrally closed if and only if its reduced class semigroup is a group and if this holds, then the C-monoid is a Krull monoid and the reduced class semigroup coincides with the usual class group of Krull monoids. We prove that a C-monoid is seminormal if and only if its reduced class semigroup is a union of groups. Based on this characterization we establish a criterion (in terms of the class semigroup) when seminormal C-monoids are half-factorial.

Keywords Krull monoids · C-monoids · Seminormal · Class semigroups · Half-factorial

Mathematics Subject Classiﬁcation 20M13 · 13A05 · 13A15 · 13F05 · 13F45

1 Introduction

A C-monoid H is a submonoid of a factorial monoid, say H ⊂ F,suchthatH × = H ∩ F× and the reduced class semigroup is finite. A commutative ring is a C-ring if its multiplicative monoid of regular elements is a C-monoid. Every C-monoid is Mori (i.e., v-noetherian), its complete integral closure H is a Krull monoid with finite class group C(H),andthe conductor (H : H) is non-trivial. Conversely, every Mori domain R with non-zero conductor f = (R : R), for which the residue class ring R/f and the class group C(R) are finite, is a C-domain ([10, Theorem 2.11.9]), and these two finiteness conditions are equivalent to being a C-domain for non-local semilocal noetherian domains ([23, Corollary 4.5]). The following result is well-known (see Sect. 2 where we gather the basics on C-monoids).

This work was supported by the Austrian Science Fund FWF, Project number P28864-N35.

B Alfred Geroldinger [email protected] https://imsc.uni-graz.at/geroldinger Qinghai Zhong [email protected] https://imsc.uni-graz.at/zhong/

1 Institute for Mathematics and Scientiﬁc Computing, NAWI Graz, University of Graz, Heinrichstraße 36, 8010 Graz, Austria 123 A. Geroldinger, Q. Zhong

Theorem A Let H be a C-monoid. Then H is completely integrally closed if and only if its reduced class semigroup is a group. If this holds, then H is a Krull monoid and the reduced class semigroup coincides with the class group of H.

The goal of the present note is the following characterization of seminormal C-monoids. Note that all seminormal C-domains are seminormal Mori domains (see [2] for a survey) and that, in particular, all seminormal orders in algebraic number ﬁelds (see [7]) are seminormal C-domains. Moreover, the seminormalization of any C-monoid is a seminormal C-monoid (Proposition 3.1).

Theorem 1.1 Let H be a C-monoid. Then H is seminormal if and only if its reduced class semigroup is a union of groups. If this holds, then we have 1. The class of every element of H is an idempotent of the class semigroup. 2. The constituent group of the smallest idempotent of the class semigroup (with respect to the Rees order) is isomorphic to the class group of the complete integral closure of H.

The finiteness of the class semigroup allows to establish a variety of arithmetical finiteness results for C-monoids (for a sample, out of many, see [10, Theorem 4.6.6] or [8]). In the special case of Krull monoids with finite class group, precise arithmetical results can be given in terms of the structure of the class group (for a survey see [25]). As a first step in this direction in the more general case of seminormal C-monoids, we establish—based on Theorem 1.1—a characterization of half-factoriality in terms of the class semigroup (Theorem 4.2).

2 Background on C-monoids

We denote by N the set of positive integers and set N0 = N ∪{0}. For rationals a, b ∈ Q, we denote by [a, b]={x ∈ Z | a ≤ x ≤ b} the discrete interval lying between a and b. All semigroups and rings in this paper are commutative and have an identity element and all homomorphisms respect identity elements. Let S be a multiplicatively written semigroup. Then S× denotes its group of invertible elements and E(S) its set of idempotents. For subsets A, B ⊂ S and an element c ∈ S we set

AB ={ab | a ∈ A, b ∈ B} and cB ={cb | b ∈ B} .

Let C be a semigroup. We use additive notation and have class groups and class semigroups in mind. Let e, f ∈ E(C).WedenotebyCe the set of all x ∈ C such that x + e = x and x + y = e for some y ∈ C.ThenCe is a group with identity element e, called the constituent group of e.Ife = f ,thenCe ∩ C f =∅.Anelementx ∈ C is contained in a subgroup of C if and only if there is some e ∈ E(C) such that x ∈ Ce. Thus C is a union of groups if and only if C = Ce . (2.1) e∈E(C) Semigroups with this property are called Clifford semigroups ([15,16]). Clearly, 0 ∈ E(C) × and C0 = C . The Rees order ≤ on E(S) is deﬁned by e ≤ f if e + f = e.IfE(S) ={0 = e0,...,en},thene0 is the largest element and e0 +···+en is the smallest idempotent in the Rees order. If C is ﬁnite, then for every a ∈ C there is an n ∈ N such that na = a +···+a ∈ E(C). 123 A characterization of seminormal C-monoids

By a monoid, we mean a cancellative semigroup. We use multiplicative notation and have × monoids of nonzero elements of domains in mind. Let H be a monoid, Hred ={aH | a ∈ H} the associated reduced monoid, and q(H) the quotient group of H.Wedenoteby • H ={x ∈ q(H) | there is an m ∈ N such that xn ∈ H for all n ≥ m} the seminormalization, • H ={x ∈ q(H) | there is an N ∈ N such that x N ∈ H} the root closure, and by • H ={x ∈ q(H) | there is a c ∈ H such that cxn ∈ H for all n ∈ N} the complete integral closure of H. Then H ⊂ H ⊂ H ⊂ H ⊂ q(H) and H is said to be • seminormal if H = H , • root closed if H = H, • completely integrally closed if H = H, • v-noetherian (or Mori) if H satisﬁes the ACC on divisorial ideals, • a Krull monoid if H is a completely integrally closed Mori monoid. × Let F be a factorial monoid, say F = F × F(P),whereF(P) is the free abelian v ( ) ∈ = ε p a monoid with basis P.Theneverya F can be written in the form a p∈P p , × where ε ∈ F and vp : F → N0 is the p-adic exponent. Let H ⊂ F be a submonoid. Two − elements y, y ∈ F are called H-equivalent (we write y ∼ y )ify−1 H ∩ F = y 1 H ∩ F or, in other words,

if for all x ∈ F, we have xy ∈ H if and only if xy ∈ H . ∈ [ ]F =[ ] H-equivalence defines a congruence relation on F and for y F,let y H y denote the congruence class of y.Then ∗ × C(H, F) ={[y]|y ∈ F} and C (H, F) ={[y]|y ∈ (F\F ) ∪{1}} are commutative semigroups with identity element [1] (introduced in [9,Section4]).C(H, F) is the class semigroup of H in F and the subsemigroup C∗(H, F) ⊂ C(H, F) is the reduced class semigroup of H in F.Wehave ∗ × × × × C(H, F) = C (H, F) ∪{[y]|y ∈ F } and {[y]|y ∈ F } =∼ F /H . (2.2) It is easy to check that either {[y]|y ∈ F×}⊂C∗(H, F) or C∗(H, F) ∩{[y]|y ∈ F×}= {[1]}. Thus ∗ C(H, F) is a union of groups if and only if C (H, F) is a union of groups (2.3) and C(H, F) is finite if and only if both C∗(H, F) and F×/H × are finite. As usual, (reduced) class semigroups and class groups will be written additively. A monoid homomorphism ∂ : H → F(P) is called a divisor theory (for H) if the following two conditions hold:

• If a, b ∈ H and ϕ(a) | ϕ(b) in F(P),thena | b in H. • For every α ∈ F(P) there are a1,...,am ∈ H such that α = gcd ϕ(a1),...,ϕ(am) . A monoid has a divisor theory if and only if it is a Krull monoid ( [10, Theorem 2.4.8]). Suppose H is a Krull monoid. Then there is an embedding of Hred into a free abelian monoid, say Hred → F(P). In this case F(P) is uniquely determined (up to isomorphism) and

C(H) = q(F(P))/q(Hred) 123 A. Geroldinger, Q. Zhong is the (divisor) class group of H which is isomorphic to the v-class group Cv(H) of H ([10, Sections 2.1 and 2.4]). A monoid H is a C-monoid if it is a submonoid of some factorial monoid F = F× ×F(P) such that H × = H ∩ F× and the reduced class semigroup is finite. We say that H is dense in F if vp(H) ⊂ N0 is a numerical monoid for all primes p of F. Suppose that H is a C-monoid. Proofs of the following facts can be found in ([10, Theorems 2.9.11 and 2.9.12]). The monoid H is v-noetherian, the conductor (H : H) =∅, and there is a factorial monoid F such that H ⊂ F is dense. Suppose that H ⊂ F is dense. Then the map v ( ) ∂ : H → F(P), defined by ∂(a) = p p a (2.4) p∈P ∗ is a divisor theory. In particular, H is a Krull monoid, Fred and hence C (H, F) are uniquely determined by H, and we call C∗(H, F) the reduced class semigroup of H. ∗ If C (H, F) is a group, then H is a Krull monoid (2.5) and every Krull monoid with finite class group is a C-monoid. Let R be a (commutative integral) domain. Then R is seminormal (completely integrally closed, Mori) if and only if its multiplicative monoid R• of nonzero elements has the respec- tive property. Moreover, R • is a Krull domain if R• is a Krull monoid, and • is a C-domain if R• is a C-monoid. Both statements generalize to rings with zero-divisors ( [12, Theorem 3.5 and Section 4]). We refer to [1,6,17,20] for C-monoids, that do not stem from ring theory, and to [18]fora more general concept.

3 A characterization of seminormality

In this section we ﬁrst show that the seminormalization and the root closure of C-monoids are still C-monoids. Since the seminormalization of a monoid is seminormal and the root closure is root closed and hence seminormal, Proposition 3.1 shows that the natural closure oper- ations, seminormalization and root closure, of C-monoids lead to seminormal C-monoids. Then we prove the characterization of seminormality, as formulated in Theorem 1.1.After that we discuss a special class of C-monoids, namely the monoid of product-one sequences over a ﬁnite group.

Proposition 3.1 Let H ⊂ F = F× × F(P) be a C-monoid. Then the canonical maps ψ : C∗(H, F) → C∗(H , F) and ψ: C∗(H, F) → C∗(H, F),

F F F F mapping [a] →[a] and [a] →[a] for every a ∈ F, are epimorphism with H H H H ψ C∗(H, F) = C∗(H , F) and ψ C∗(H, F) = C∗(H, F). Moreover, H ⊂ F is a seminormal C-monoid and H ⊂ F is a root closed C-monoid. If every class of C∗(H, F) contains an element from P, then every class of C∗(H , F) and every class of C∗(H, F) contains an element from P.

Proof We have H ⊂ H ⊂ H ⊂ F, H is seminormal, H is root closed, H × = H ∩ F×, (H )× ⊂ H ∩ F×,and(H)× ⊂ H ∩ F×.Leta ∈ H ∩ F×.Thenak ∈ H ∩ F× = H × 123 A characterization of seminormal C-monoids for some k ∈ N and hence there exists b ∈ H × ⊂ H such that akb = 1. Since ak−1b ∈ H , we obtain that a ∈ (H )×. Therefore (H )× = H ∩ F×.Leta ∈ H ∩ F×.Thenak ∈ H ∩ F× = H × for some k ∈ N and hence there exists b ∈ H × ⊂ H such that akb = 1. Since ak−1b ∈ H, we obtain that a ∈ (H)×. Therefore (H)× = H ∩ F×. ψ , ∈ [ ]F =[]F ∈ To verify that is well-deﬁned, let a b F with a H b H .Foranx F with ∈ ∈ [ ]F =[]F ax H we have to verify that bx H . Then this implies that a H b H . In fact, since ax ∈ H , there exists N ∈ N such that for all n ≥ N,wehave(ax)n ∈ H.Since [ ]F =[]F [ n]F =[ n]F n n ∈ n n ∈ a H b H , it follows that a H b H whence a x H implies that b x H. Therefore for all n ≥ N,wehave(bx)n ∈ H which implies that bx ∈ H . ψ , ∈ [ ]F =[ ]F ∈ To verify that is well-deﬁned, let a b F with a H b H and let x F such that ∈ ∈ ∈ N ( )N ∈ [ ]F =[]F ax H.Sinceax H, there exists N such that ax H.Since a H b H , [ N ]F =[ N ]F N N ∈ N N ∈ it follows that a H b H whence a x H implies that b x H. Therefore bx ∈ H. Obviously, ψ and ψ are epimorphisms with

ψ (C∗( , )) = ψ ({[ ]F | ∈ \ × ∪{ }}) ={[ ]F | ∈ \ × ∪{ }} = C∗( , ) H F a H a F F 1 a H a F F 1 H F and ψ( C∗(H, F)) = ψ( {[a]F | a ∈ F\F× ∪{ }}) ={[a]F | a ∈ F\F× ∪{ }} = C∗(H, F). H 1 H 1 Thus H ⊂ F is a seminormal C-monoid and H ⊂ F is a root closed C-monoid.

Proof of Theorem 1.1 By [10, Theorem 2.9.11.4], we may choose a special factorial monoid F such that H ⊂ F is a dense C-monoid. This special choice will turn out to be useful in the × proof of part 3. Since H is a Krull monoid, we have H = H × H0,whereH0 is a reduced × Krull monoid. If the embedding H0 → F(P) is a divisor theory, then H ⊂ F = H ×F(P) is a dense C-monoid. In particular, we have H × = H ∩ F× and C∗ = C∗(H, F) is ﬁnite. Furthermore, there are α ∈ N and a subgroup V ⊂ F× such that the following properties hold ([10, Proposition 2.8.11 and Deﬁnition 2.9.5]):

P1. H × ⊂ V ,(F× : V ) | α, V (H\H ×) ⊂ H, P2. q2α F ∩ H = qα(qα F ∩ H) for all q ∈ F\F× .

In particular, if p ∈ P and a ∈ pα F,thena ∈ H if and only if pαa ∈ H. 1. First we suppose that H is seminormal. We have to show that C∗ is a union of groups. ∈ \ × = k1 · ...· kt ∈ × ,..., ∈ We choose an element a F F ,saya p1 pt ,where F , p1 pt P, t, k ,...,k ∈ N.SinceC∗ is ﬁnite, there is an n ∈ N such that [an] is an idempotent element 1 t of C∗. Since for every multiple n of n, [an ] is an idempotent element of C∗,wemayassume without restriction that n is a multiple of α. We assert that an+1 ∼ a. Clearly, this implies that [a] lies in a cyclic subgroup of C∗. In order to show that an+1 ∼ a, we choose an element b ∈ F. First, suppose that ab ∈ H. ≥ α ( )r ∈ ∩ α · ...· α Then for every r ,wehave ab H p1 pt F. It follows by Properties (P1) α n α × n = ( ) α ∈ ( )r ∈ \ ∈[ , ] and (P2) that V and p j ab H H for all j 1 t . Using Property (P2) repeatedly we infer that

( /α)α ( /α)α ( n+1 )r = rn( )r = rn rk1 n · ...· rkt n ( )r ∈ ( \ ×) ⊂ . a b a ab p1 pt ab V H H H Since H is seminormal, we obtain an+1b ∈ H. Conversely, suppose that ban+1 ∈ H.Wehavetoverifythatba ∈ H.Todosoweclaim that

(ba)man ∈ H for all m ∈ N , (3.1) 123 A. Geroldinger, Q. Zhong and we proceed by induction on m.Ifm = 1, then this holds by assumption. Suppose the claim holds for m ∈ N. Then, by the induction hypothesis,

(ba)m+1an ∼ (ba)m+1a2n = (ba)man ban+1 ∈ H whence (ba)m+1an ∈ H.Using(3.1) with m = n we infer that (ba)n = bnan ∼ bna2n = (ba)nan ∈ H whence (ba)n ∈ H. Thus we obtain that

(ba)n+1 = (bn+1a)an ∼ (bn+1a)a2n = (ba)n ban+1 ∈ H whence (ba)n+1 ∈ H.Thisimpliesthat(ba) ∈ H for all sufficiently large ∈ N.SinceH is seminormal, it follows that ba ∈ H. 2. Now we suppose that C∗ is a union of groups. Then, by (2.2), the class semigroup C(H, F) is a union of groups. In order to show that H is seminormal, let a ∈ q(H) be given such that an ∈ H for all n ≥ N for some N ∈ N.SinceF is factorial, it is seminormal whence H ⊂ F and q(H) ⊂ q(F) imply that a ∈ F.Ifa ∈ F×,thena = a N+1(a N )−1 ∈ H. Suppose a ∈ F\F×.Since[a]∈C∗ and C∗ is finite, we obtain that [a] ⊂ C∗ is a finite ∗ N t+1 cyclic group. If N0 ∈ N is the order of [a] in the cyclic group [a] ⊂ C ,thena ∼ a 0 N N+1 for all t ∈ N.SinceN0 N + 1 > N, we obtain that a 0 ∈ H and hence a ∈ H. ∗ 3. Suppose that H is seminormal and set E(C ) ={e0,...,en},wheren ∈ N0, e0 =[1], e +···+e = e i ∈[ , n] C∗ = C∗ C∗ and 0 n n.For 0 ,weset i ei .Since is a union of groups, (2.1) implies that n C∗ = C∗ . i i=0 ∈[ , ] C∗ ={ ∈ C∗ | = ∈ N} C∗ + ∗ = ∗ For every i 0 n ,wehave i g mg ei for some m and i Cn Cn . For the class group of the Krull monoid H,wehave C(H) = q(F(P))/q(H0) = q(F)/q(H) = q(F)/q(H). × × Since H0 → F(P) is a divisor theory, the inclusion H = H × H0 ⊂ F = H × F(P) is saturated whence H = q(H) ∩ F = q(H) ∩ F ([10, Corollary 2.4.3.2]). 3.(a) Let a ∈ H.Ifa ∈ H ×,then[a]=[1] is an idempotent of C∗. Suppose a ∈ H\H ×. We have to show that a ∼ a2 and to do so we choose some b ∈ F.Ifab ∈ H, then obviously a2b ∈ H. Conversely, suppose that a2b ∈ H.SinceC∗ is a finite Clifford semigroup, there exists an m ∈ N such that am+1 ∼ a.Thenab ∼ am+1b = am−1a2b ∈ H which implies ab ∈ H. 3.(b) By [10, Proposition 2.8.7.1], the map : C∗ → C(H) given by ([a]) = aq(H) ∗ = |C∗ : C → C( ) is an epimorphism. Therefore n n n H is a group homomorphism, and it remains to show that it is bijective. × ∗ Let b ∈ F\F such that [b]=en.Sinceen is the identity of C , n(en) is the identity n element of C(H) whence bq(H) = n(en) = q(H) and b ∈ q(H) ∩ F = H. Then for ∈ [ ]=[ ]+[ ]∈C∗ ([ ]) = q( ) = q( ) every a F we have ba b a n and ba ba H a H . Thus n is surjective. ∈ \ × [ ]∈C∗ ([ ]) C( ) Let a F F with a n such that n a is the identity element of H .Then aq(H) = q(H) and a ∈ q(H) ∩ F = H whence there exists c ∈ H such that ca ∈ H. Thus [ ] [ ] [ ] C∗ ∗ + ∗ = ∗ 3.(a) implies that ca and c are idempotents. Since a and en are in n and Ci Cn Cn 123 A characterization of seminormal C-monoids

∈[ , ] [ ]=[ ]+[ ] [ ]+ C∗ [ ]=[ ]+[ ] for all i 0 n , it follows that ca c a and c en are in n .Since ca ca ca , it follows that [ca]=en and similarly [c]+en = en. Thus we obtain that

[a]=[a]+en =[a]+[c]+en =[ca]+en = en , which implies that n is injective. In our next remark and also in Sect. 4, we need the concept of monoids of product-one sequences over groups. Let G be a finite group and F(G) be the free abelian monoid with basis G.AnelementS = g1 · ...· g ∈ F(G) is said to be a product-one sequence (over G) if its terms can be ordered such that their product equals 1G , the identity element of the group G. The monoid B(G) of all product-one sequences over G is a finitely generated C-monoid, and it is a Krull monoid if and only if G is abelian ([6, Theorem 3.2]). Remark 3.2 1. The main statement of Theorem 1.1 was proved first for the monoid of product-one sequences over a finite group. Let G be a finite group and G be its commutator subgroup. JunSeokOhshowedthatB(G) is seminormal if and only if |G |≤2 if and only if its class semigroup C∗(B(G), F(G)) is a union of groups ([21, Corollary 3.12]). 2. Let H ⊂ F be a C-monoid with all conventions as in Theorem 1.1.Ifa ∈ H,then Theorem 1.1.1 implies that the element [a]∈C(H, F) is idempotent. In case H = B(G), the converse holds and this fact was used in the characterization when B(G) is seminormal. However, the converse does not hold true for general C-monoids as the next example shows. 3. Let F = F(P) with P ={p1, p2} and =[ , 2k | ≥ ]={ , }∪{ 2k | ≥ }∪{ t s | ≥ , ≥ }⊂ . H p1 p2 p1 p2 k 0 1 p1 p2 p1 p2 k 0 p1 p2 t 0 s 2 F ( : 2) = ( : 4) = \{ , } [ 2]=[ 4]∈C( , ) Then H p1 H p1 H 1 p1 p2 which implies that p1 p1 H F . Thus [ 2] 2 ∈/ p1 is an idempotent but p1 H.

4 A characterization of half-factoriality

Let H be a monoid and A(H) the set of atoms (irreducible elements) of H.Ifanelement a ∈ H has a factorization of the form a = u1 · ...· uk,wherek ∈ N and u1,...,uk ∈ A(H), then k is called a factorization length of a.ThesetLH (a) = L(a) of all factorization lengths is called the set of lengths of a, and for simplicity we set L(a) ={0} for a ∈ H ×.IfH is v-noetherian (which holds true of all C-monoids), then every a ∈ H has a factorization into atoms and all sets of lengths are finite ( [10, Theorem 2.2.9]). For a finite set L = {m0,...,mk }⊂Z,wherek ∈ N0 and m0 < ···< mk, (L) ={mi − mi−1 | i ∈[1, k]} is called the set of distances of L.Then L(H) ={L(a) | a ∈ H} resp. (H) = (L) L∈L(H) is called the system of sets of lengths resp. the setofdistancesof H. The monoid H is half-factorial if (H) =∅(equivalently, |L|=1forallL ∈ L(H)). Half-factoriality has always been a central topic in factorization theory (e.g., [3–5,19,22, 24]). In 1960 Carlitz proved that a ring of integers in an algebraic number field is half-factorial if and only if the class group has at most two elements. This result (which has a simple proof nowadays) carries over to Krull monoids. Indeed, a Krull monoid with class group G,that 123 A. Geroldinger, Q. Zhong has a prime divisor in each class, is half-factorial if and only if |G|≤2 (the assumption on the distribution of prime divisors is crucial; we refer to [14, Section 5] for background if this assumption fails). We will need this result and the involved machinery in our study of half-factoriality for C-monoids. Thus, we recall that a monoid homomorphism θ : H → B is a transfer homomorphism if the following conditions hold: (T 1) B = θ(H)B× and θ −1(B×) = H ×. (T 2) If u ∈ H, b, c ∈ B and θ(u) = bc, then there exist v, w ∈ H such that u = vw, θ(v)B× = bB× and θ(w)B× = cB×. If θ : H → B is a transfer homomorphism, then L(H) = L(B) ([10, Proposition 3.2.3]) whence H is half-factorial if and only if B is half-factorial. All generalizations of Carlitz’s result beyond the setting of Krull monoids have turned out to be surprisingly difficult. We mention two results valid for special classes of C-domains. There is a characterization of half-factoriality for orders in quadratic number fields in number theoretic terms ([10, Theorem 3.7.15]) and for a class of seminormal weakly Krull domains (including seminormal orders in number fields) in algebraic terms involving the v-class group and extension properties of prime divisorial ideals (see [11, Theorem 6.2] and [13]). In this section we establish a characterization of half-factoriality in terms of the class semigroup, that is valid for all seminormal C-monoids and based on Theorem 1.1. Although it is not difficult to show that the set of distances is finite for all C-monoids ([10, Theorem 3.3.4]), a characterization of half-factoriality turns out to be quite involved compared with the simpleness of the result for Krull monoids. But this difference in complexity stems from the fact that the structure of the class semigroup of a C-monoid H can be much more intricate than the structure of the class group C(H) of its complete integral closure. We provide an explicit example of a half-factorial seminormal C-monoid (Example 4.3) and we also refer to the explicit examples of class semigroups given in [20, Section 4]. We introduce our notation which remains valid throughout the rest of this section. Let × H ⊂ F = F × F(P) be a dense seminormal C-monoid with E(C) ={e0 =[1],...,en}, where n ∈ N and e0 +···+en = en. For a subset T ⊂ F,wedefine

CT (H, F) ={[y]|y ∈ T }⊂C(H, F). We use the abbreviations C = C(H, F), C∗ = C∗(H, F), C = C (H, F), C∗ = C∗ , H H i ei and C = C ∈[ , ] , i ei for all i 0 n whence n n E(C∗) = E(C), C = C , C∗ = C∗, C = C∗ ∈[ , ], i i i i for all i 1 n and i=0 i=0 × ∗ C = C = C × ( , ) ∪ C , 0 F H F 0 Proposition 4.1 Let H ⊂ F = F× × F(P) be a dense seminormal C-monoid (with all notations as above) and suppose that every class of C∗ contains an element from P. For ∗ × every i ∈[0, n],letPi ={p ∈ P |[p]∈C },Fi = F × F(Pi ),Hi = Fi ∩ H, and ∗ i × ϕ : C × ( , ) → C ϕ ([]) =[]+ ∈ let i F H F i be deﬁned by i ei for all F . Then for every k ∈[0, n],Hk ⊂ Fk is a seminormal C-monoid and the following statements hold: × 1. If ek ∈/ CH ,thenHk = H .Ifek ∈ CH ,then

∼ Ck, if ϕk is injective, C(Hk, Fk) = CF× (H, F) ∪ Ck, if ϕk is not injective. 123 A characterization of seminormal C-monoids

∼ In particular, C(H0, F0) = C0 and if C(Hk, Fk) is a group, then Hk is a Krull monoid. ∗ ∈ C θ : → B C /ϕ (C × ( , )) 2. If ek H , then there is a transfer homomorphism k Hk k k F H F .

Proof Let k ∈[0, n].By[10, Proposition 2.9.9], Hk ⊂ Fk is a C-monoid and it is a divisor- closed submonoid of H.SinceHk is a divisor-closed submonoid of the seminormal monoid H, it is seminormal (this is easy to check; for details see [11, Lemma 3.2]). By [10, Lemma F Fk 2.8.4.5], ψk : CF (H, F) → C(Hk, Fk ),definedbyψk ([a] ) =[a] for a ∈ Fk,isan k H ∗ Hk epimorphism. By definition, we have CF (H, F) = CF× (H, F) ∪ C = CF× (H, F) ∪ Ck. k × k 1. First suppose ek ∈/ CH . It is clear that H ⊂ H ∩ Fk = Hk. Assume to the contrary that × × × × there exists an element a ∈ Hk\H .Ifa ∈ F ,thena ∈ F ∩ H = H , a contradiction. ∈ \ × [ ]F ∈ C [ ]F = ∈ C If a Fk F ,then a H k and again Theorem 1.1.1 implies that a H ek H ,a contradiction. Now we suppose that ek ∈ CH . ϕ ψ | : C → C( , ) 1.(a) Suppose that k is injective. We want to show that k Ck k Hk Fk is bijective. ψ | ∈ [ ]F = ∈ × First, we show that k Ck is surjective. Let p Pk such that p H ek and F . Then p ∈ H . It suffices to prove []Fk =[ p]Fk . k Hk Hk ∈ ∈ ∈ ∈ [ ]F =[ ]F + Let a Fk.If a Hk,then pa Hk. Suppose pa Hk.Then pa H a H = [ ]F ∈ C [ ]F = ∈ ∩ = ek ek.If a H k,then a H ek and hence a H Fk Hk . Suppose [ ]F ∈ C × ( , ) ϕ [ ]F =[ ]F ∈ a H F H F .Since k is injective, we obtain a H 1 H whence a Hk . ψ | , ∈ [ ]F =[ ]F In order to show that k Ck is injective, let p1 p2 Pk be given such that p1 H p2 H . C∗ ∈ [ ]F = Since every class of contains an element from P,thereisap Pk such that p H −[p ]F .Thenpp ∈ H and pp ∈/ H which implies that [p ]Fk =[p ]Fk . 1 H 1 k 2 k 1 Hk 2 Hk 1.(b) Suppose that ϕk is not injective. It suffices to prove ψk is injective. Let a1, a2 ∈ Fk such that [a ]F =[a ]F .Wehavetoshowthat[a ]Fk =[a ]Fk and to do so we distinguish 1 H 2 H 1 Hk 2 Hk three cases. × [ ]F , [ ]F ∈ C × ( , ) ∈ []F =−[ ]F If a1 H a2 H F H F , then there exists an F such that H a1 H . Therefore a ∈ H and a ∈/ H which imply [a ]Fk =[a ]Fk . 1 k 2 k 1 Hk 2 Hk [ ]F , [ ]F ∈ C ∈ [ ]F =−[ ]F If a1 H a2 H k, then there exists an a Fk such that a H a1 H . Therefore aa ∈ H and aa ∈/ H which imply [a ]Fk =[a ]Fk . 1 k 2 k 1 Hk 2 Hk [ ]F ∈ C [ ]F ∈ C × ( , ) By symmetry, it remains to consider the case where a1 H k and a2 H F H F . ∈ × []F =−[ ]F ∈ Then there exists F such that H a2 H . Therefore a2 Hk. Assume to the contrary that [a ]Fk =[a ]Fk .Thena ∈ H .Sinceϕ is not injective, there exists η ∈ F× 1 Hk 2 Hk 1 k k [η]F =[]F [η]F + = [η ]= η ∈ such that H 1 H and H ek ek.Then a1 ek whence a1 Hk and F F F ηa2 ∈ Hk.But[ηa2] =[η] =[1] , a contradiction. H H H ∼ 1.(c) Since CF× (H, F) ∪ C0 = C0,wehaveC(H0, F0) = C0.IfC(Hk, Fk) is a group, then Hk is a Krull monoid by (2.3)and(2.5). ∗ 2. Suppose that ek ∈ CH .Forg ∈ C ,wesetg = g + ϕk(CF× (H, F)) ∈ ∗ k C /ϕ (C × ( , )) k k F H F and we define ∗ ∗ θ : −→ F(C /ϕ (C × ( , ))) Fk k k F H F v ( ) v ( ) p a = p a −→ [ ]F . a p p H p∈Pk p∈Pk

∗ ∗ θ ( ) = B(C /ϕ (C × ( , ))) = ∩ = ∈ | First, we show that Hk k k F H F .NoteHk H Fk x Fk × F ∗ ∗ vp(a) [x] ∈{e , e } .Ifa ∈ H ,thenθ (a) = 1F(C /ϕ (C × ( , ))).Ifa = p ∈ H 0 k k k k F H F p∈Pk × H \H , then the sum of the elements of θ ∗(a) equals v (a)[p]F = [a]F = e , k k p∈Pk p H H k 123 A. Geroldinger, Q. Zhong

∗ ∗ which is the zero element of the group C /ϕk (CF× (H, F)). Thus we obtain that θ (Hk) ⊂ ∗ k B(C /ϕk (CF× (H, F))) and in order to verify equality, we choose an S = g1 · ...· g ∈ k∗ ∗ B(C /ϕ (C × ( , ))) ,..., ∈ C ,..., ∈ k k F H F ,whereg1 g k . By assumption, there exist p1 p Pk [ ]F = ∈[ , ] such that pi H gi for all i 1 .Since

F F F ∗ [p · ...· p ] = [p ] +···+[p ] = g +···+g = 0C /ϕ (C × ( , )) , 1 H 1 H H 1 k k F H F × [ ·...· ]F ∈ ϕ (C × ( , )) ∈ ϕ ([]F ) = we infer that p1 p H k F H F . Thus there is an F such that k H F F −[p1 ·...· p ] . So it follows that [ p1 ·...· p ] = ek whence p1 ·...· p ∈ H ∩ Fk = Hk ∗ H H and θ ( p1 · ...· p ) = S. ∗ ∗ θ = θ | : → B(C /ϕ (C × ( , ))) Secondly, we show Hk Hk k k F H F is a transfer homomorphism. Clearly, B(·) is reduced and (T1) holds. In order to verify (T2),leta ∈ Hk ∗ and b, c ∈ B(C /ϕk(CF× (H, F)))\{1} such that θ(a) = bc.Weseta = p1 · ... · p , × k where ∈ F , ∈ N0, p1,...,p ∈ Pk and, after renumbering if necessary, we assume = [ ]F · ...· [ ]F = [ ]F · ...· [ ]F ∈[ , − ] b p1 H ps H and c ps+1 H p H ,wheres 1 1 . Thus there exists ∈ × ϕ ([ ]F ) =[ ]F + =−[ · ...· ]F an 1 F such that k 1 H 1 H ek p1 ps H and hence [−1]F + =[]F + −[ ]F + =−[ · ...· ]F +[ · ...· ]F 1 H ek H ek 1 H ek p1 p H p1 ps H =−[ · ...· ]F . ps+1 p H = · ...· ∈ v = −1 · ...· ∈ = v θ( ) = It follows that u 1 p1 ps Hk , 1 ps+1 p Hk, a u , u b, and θ(v) = c.

Theorem 4.2 Let H ⊂ F = F××F(P) be a dense seminormal C-monoid (with all notations as above) and suppose that every class of C∗ contains an element from P. For every i ∈ ∗ × [0, n],letϕi : CF× (H, F) → C be deﬁned by ϕi ([]) =[]+ei for all ∈ F and set ∗ i ∗ ∗ ∗ C = C /ϕ (C × ( , )) , ∈[ , ] φ : C → C + C i i i F H F . For distinct i j 0 n ,let i, j i i j be deﬁned by φ ( ) = + ∈ C∗ i, j gi gi e j for all gi i . Then H is half-factorial if and only if the following properties hold: |C |≤ P1. n 2. P2. For every i ∈[0, n], we have

− ϕi (CF× (H, F)), ei ∈ CH φ 1(ϕ (C × ( , ))) = i,n n F H F Ci , ei ∈/ CH

P3. For distinct i, j ∈[0, n],ifei , e j ∈ CH ,thenker(φi, j ◦ ϕi ) = ker(ϕi ) + ker(ϕ j ). E(C∗)\C , , ∈[, ] , ∈ C P4. H is additively closed and for distinct i1 i2 j 0 n ,ifei1 ei2 H and ∈ E(C∗)\C + + ∈ C + ∈ C + ∈ C e j H such that ei1 ei2 e j H ,thenei1 e j H or ei2 e j H .

∗ ∗ Proof In order to show that en ∈ CH , we choose a p ∈ P such that [p ]=en.Since H ⊂ F is dense, there exists an a ∈ H such that p∗ | a,saya = p∗b with b ∈ F.By ∗ Theorem 1.1, [a] is an idempotent and since [a]=[p ]+[b]=en +[b]∈Cn, it follows that en =[a]∈CH . 1. We suppose that H is half-factorial and verify properties (P1) - (P4).

={ ∈ |[ ]∈C∗} = × × F( ) = ∩ (P1). Let Pn p P p n , Fn F Pn ,andHn H Fn.SinceHn is a divisor closed submonoid of H, Hn is half-factorial. By Proposition 4.1.3, there is a θ : → B( ) B( ) transfer homomorphism Hn Cn . Thus Cn is half-factorial which implies |C |≤ that n 2([10, Theorem 3.4.11.5]). 123 A characterization of seminormal C-monoids

(P2). Let i ∈[0, n] and ei ∈ CH . Clearly, ei + en = en and φi,n(CF× (H, F) + ei ) = ∗ CF× (H, F) + en. Assume to the contrary that there exists gi ∈ C with gi ∈/ i × ϕi (CF× (H, F)) such that φi,n(gi ) ∈ CF× (H, F) + en, i.e. there exists ∈ F such that gi + en =[]+en.Letp1, p2, q ∈ P such that [p1]=gi , [p2]=−gi , −1 and [q]=en.Then[ p1q]=[ p2q]=[q]=en and [p1 p2]=ei .We −1 claim that p1q,p2q, p1 p2, q ∈ A(H). Clearly, the elements lie in H and, for example, assume to the contrary that p1 p2 is not an atom. Then there is an × −1 η ∈ F such that η p1,η p2 ∈ H.Then [η p1]=[η]+[p1]=ei whence −1 gi =[p1]=[η ]+ei ∈ ϕi CF× (H, F) , a contradiction. Thus we obtain that 2 −1 (p1 p2)(q) = ( p1q)( p2q), a contradiction to the half-factoriality of H. ∗ Suppose ei ∈ E(C )\CH . Assume to the contrary that there exists gi ∈ Ci such that φi,n(gi ) = gi + en ∈/ CF× (H, F) + en.Letp1, p2, p3 ∈ P such that [p1]=gi , [ ]= [ ]=− + ( ) = ( ) p2 en,and p3 gi en,andletord gi ordCi gi denote the order of gi ( ) C [ ord gi ]=[ ]=[ ]= ∈ C in the group i .Then p1 p2 p2 p1 p3 en H which implies that (g ) ( ) (g ) (g ) ord i , , ∈ A( ) ( )ord gi = ( ord i )( ord i ) p1 p2 p2 p1 p3 H . Therefore p1 p3 p2 p1 p2 p3 . ( ) ( ) ord gi ∈ ∈ × ∈/ ( )/∈ L ( ord gi ) Since p3 H and for any F , p3 H,weobtainord gi H p3 , a contradiction to the half-factoriality of H. (P3). Let i, j ∈[0, n] be distinct and ei , e j ∈ CH .Sinceφi, j ◦ ϕi = φ j,i ◦ ϕ j ,wehave ker(φi, j ◦ ϕi ) ⊃ ker(ϕi ) and ker(φi, j ◦ ϕi ) ⊃ ker(ϕ j ) which imply ker(φi, j ◦ ϕi ) ⊃ × ker(ϕi ) + ker(ϕ j ).Let ∈ F such that φi, j (ϕi ([])) = ei + e j .If[]+ei = ei or []+e j = e j ,then[]=[]+[1]∈ker(ϕi )+ker(ϕ j ). Suppose []+ei = ei and []+ e j = e j .Letp1, p2, p3 ∈ P such that [p1]=[]+ei , [p2]=e j ,and[p3]=ei + e j . −1 −1 Since p2, p3,p3, p1 ∈ A(H), the equation (p1 p2)p3 = ( p1)( p3)p2 implies × −1 that p1 p2 ∈/ A(H). Then there exists δ ∈ F such that δ p1 ∈ H and δ p2 ∈ H −1 whence [δ p1]=[δ]+[]+ei = ei and [δ p2]=−[δ]+e j = e j . It follows that [δ]+[]∈ker(ϕi ), −[δ]∈ker(ϕ j ),and[]=[δ]+[]−[δ]∈ker(ϕi ) + ker(ϕ j ). ∗ (P4). Assume to the contrary that there exist f1, f2 ∈ E(C )\CH such that f1 + f2 ∈ CH . , ∈ [ ]= [ ]= [ ]=[ 2 ]=[ 2]= Let q1 q2 P such that q1 f1 and q2 f2.Then q1q2 q1 q2 q1q2 + ∈ C , 2 , 2 ∈ A( ) ( )3 = ( 2 )( 2) f1 f2 H , q1q2 q1 q2 q1q2 H ,and q1q2 q1 q2 q1q2 , a contradiction to the half-factoriality of H. , , ∈[ , ] ∈ E(C∗)\C , , + + ∈ C Let i1 i2 j 0 n such that e j H and ei1 ei2 ei1 ei2 e j H . Assume + , + ∈ E(C∗)\C , , , ∈ to the contrary that ei1 e j ei2 e j H .Letp p1 p2 p3 P such that [ ]= + , [ ]= , [ ]= , [ ]= , , , , ∈ p ei1 ei2 p1 ei1 p2 ei2 p3 e j .Thenp p1 p2 pp3 p1 p2 p3 A(H) and (p)(p1 p2 p3) = (p1)(p2)(pp3), a contradiction to the half-factoriality of H. 2. We suppose that (P1) - (P4) hold and prove that H is half-factorial. We start with the following three assertions. ∈[ , ] ∈ C |C |≤ A1. If i 0 n and ei H ,then i 2. − , ∈[ , ] , ∈ C φ 1(C × ( , ) + + ) = C × ( , ) + A2. If i j 0 n and ei e j H ,then i, j F H F ei e j F H F ei . − , ∈[ , ] ∈/ C ∈ C + ∈ C φ 1(C × ( , ) + A3. If i j 0 n , ei H ,ande j H such that ei e j H ,then i, j F H F ei + e j ) = Ci . ∈[ , ] ∈ C φ : C → C Proof of A1.Leti 0 n and ei H . By (P2), there is a monomorphism , n i ∗n i φ ( + ϕ (C × ( , ))) = + + ϕ (C × ( , )) ∈ C deﬁned by i,n h i F H F h en n F H F for all h i . It follows |C |≤ by (P1) that i 2. Proof of A2.Leti, j ∈[0, n] and ei , e j ∈ CH ,sayei + e j = ek with k ∈[0, n],and note that φi,n = φk,n ◦ φi,k and φi, j = φi,k. Then (P2) implies that CF× (H, F) + ei = − − − − φ 1(C × ( , ) + ) = φ 1(φ 1(C × ( , ) + )) = φ 1(C × ( , ) + ) i,n F H F en i,k k,n F H F en i, j F H F ek . 123 A. Geroldinger, Q. Zhong

∗ Proof of A3.Leti, j ∈[0, n], ei ∈ E(C )\CH ,ande j ∈ CH such that ei + e j ∈ CH ,say ei + e j = ek with k ∈[0, n], and again we note that φi,n = φk,n ◦ φi,k and φi, j = φi,k. − − − C = φ 1(C × ( , ) + ) = φ 1(φ 1(C × ( , ) + )) = Then (P2) implies that i i,n F H F en i,k k,n F H F en − φ 1(C × ( , ) + ) i, j F H F ek . Now we set I ={i ∈[0, n]|ei ∈ CH } and we deﬁne ∗ ∗ ∗ 1 = ϕ C × ( , ) , 2 = C \ϕ (C × ( , )) , 3 = C \ C . G i F H F G i i F H F and G i i∈I i∈I i∈I × For every a = p1 · ...· p ∈ F,where ∈ F , ∈ N0,andp1,...,p ∈ P,wedeﬁne

1 2 l1(a) =|{j ∈[1, ]|[p j ]∈G }| , l2(a) =|{k ∈[1, ]|[pk ]∈G }| , and 1 l(a) = l (a) + l (a). 1 2 2 In order to prove that H is half-factorial, it is sufﬁcient to prove the following assertion. A4. If a is an atom of H,thenl(a) = 1. × Proof of A4.Leta = p1 ·...· p ∈ F\F be an atom of H. Assume to the contrary that l( ) = ∈[ , ] [ ]∈ 3 [ ]∈C ∈ E(C∗)\C a 0. Then, for every i 1 ,wehave pi G ,say pi fi with fi H . [ ]=[]+[ ]+···+[ ]∈C [ ]∈C × ( , )+C +···+C ⊂ Clearly,wehave a p1 p [a] and a F H F f1 f C +···+ =[ ] f1+···+ f whence f1 f a , a contradiction to (P4). Assume to the contrary that l(a) ≥ 2 and distinguish three cases. CASE 1: l1(a) ≥ 2. 1 After renumbering if necessary, we may assume that [p1], [p2]∈G .Thenthereexists × 1,2 ∈ F and ei , e j ∈ CH such that [p1]=[1]+ei , [p2]=[2]+e j .Wesetg = [ p3 · ...· p ]∈Ck and [a]=er ∈ CH for some k, r ∈[0, n].Then

=[ ]+ +[ ]+ + ∈ C × ( , ) + C + C + C ⊂ C er 1 ei 2 e j g F H F i j k ei +e j +ek whence ei + e j + ek = er ∈ CH . By (P4), we obtain that ei + ek ∈ CH or e j + ek ∈ CH ,say e j + ek = et ∈ CH for some t ∈[0, n]. Since [1]+[p2]+g ∈ Ct and

φt,i ([1]+[p2]+g) =[1]+[p2]+g + ei =[p1]+[p2]+g =[ p1 · ...· p ]=er

= ei + et ∈ CF× (H, F) + ei + et , we obtain [1]+[p2]+g ∈ CF× (H, F)+et by A2. It follows that [ p2 ·...· p ]=[p2]+g ∈ × CF× (H, F) + et whence there exists 3 ∈ F such that [ p2 · ...· p ]=[3]+et .Since

ei + et = er =[a]=[p1]+[ p2 · ...· p ]=[1]+ei +[3]+et , × we infer that [1]+[3]∈ker(φi,t ◦ϕi ).By(P3),thereexistsδ ∈ F such that [δ]∈ker(ϕi ) −1 and [13δ ]∈ker(ϕt ). Thus we obtain that [δ−1 ]=[δ]−[ ]+[ ]=[δ]+ = 1 p1 1 p1 ei ei [ δ−1 · ...· ]=[ δ−1−1 · ...· ]=[ δ−1]−[ ] 1 p2 p 1 3 3 p2 p 1 3 3 −1 +[ p2 · ...· p ]=[12δ ]+et = et . δ−1 ∈ \ × δ−1 ·...· ∈ \ × = (δ−1 )( δ−1 ·...· ) Thus 1 p1 H H , 1 p2 p H H ,anda 1 p1 1 p2 p , a contradiction to a ∈ A(H). 123 A characterization of seminormal C-monoids

CASE 2: l1(a) = 1andl2(a) ≥ 2. 1 2 After renumbering if necessary, we may assume that [p1]∈G ,and[p2], [p3]∈G . ∗ ∗ , , ∈[, ] [ ]∈C \ϕ (C × ( , ) [ ]∈C \ϕ (C × ( , ) There are i j k 0 n such that p2 i i F H F , p3 j j F H F , and ek = ei + e j .ByA2 we infer that

φi,k([p2]) =[p2]+ek ∈/ CF× (H, F) + ei + ek = ϕk(CF× (H, F)) and

φi,k([p3]) =[p3]+ek ∈/ CF× (H, F) + ei + ek = ϕk(CF× (H, F)) . ∗ [ ]+ [ ]+ C \ϕ (C × ( , )) |C |≤ Since p2 ek and p3 ek are in k k F H F and since, by A1, k 2, it 1 follows that [p2]+[p3]=([p2]+ek) + ([p3]+ek) ∈ ϕk(CF× (H, F)) ⊂ G . We choose a q ∈ P such that [q]=[p2]+[p3].Thenb = p1qp4 ·...· p ∈ A(H) with l(b) = l(a) ≥ 2. Now we are back to CASE 1. CASE 3: l1(a) = 0andl2(a) ≥ 4. 2 After renumbering if necessary, we may assume that [p1], [p2], [p3], [p4]∈G . Arguing 1 as in CASE 2 we infer that [p1]+[p2], [p3]+[p4]∈G .Letq1, q2 ∈ P such that [q1]=[p1]+[p2] and [q2]=[p3]+[p4].Thenb = q1q2 p4 · ...· p ∈ A(H) with l(b) = l(a) ≥ 2 whence we are back to CASE 1.

The following example shows that both the number and the size of the constituent groups of the reduced class semigroup of a half-factorial seminormal C-monoid can be arbitrarily large.

Example 4.3 Let C be a Clifford semigroup with n E( ) ={ , ,..., }, = , = = ⊕ Z/ Z , C e0 e1 en C Ci and Cei Ci Gi 2 i=0 where n ∈ N, i, j ∈[0, n], Ci + e j = C j whenever i < j,andG0 ··· Gn ={1} are abelian groups. Let × F = F × F(C) and B ={S ∈ F | ι() + σ(S) ∈ E(C)} , × × where F = G0, ι: F → C0 is a monomorphism, and σ : F(C) → C is the sum function, which is deﬁned as σ(g1 · ...· g ) = g1 +···+g .ThenB ⊂ F is a half-factorial dense seminormal C-monoid such that every class of C(B, F) contains an element from C, ∗ ∼ ∼ × ∗ C (B, F) = C(B, F) = C, CF× (B, F) = F , and CB (B, F) = E C (B, F) . × Proof We start with the following four assertions. Let , 1,2 ∈ F , S, S1, S2 ∈ F(C),and consider the epimorphism × × λ: F × F(C) → C deﬁned by λ(S) = ι() + σ(S) for all S ∈ F × F(C).

A1. 1 S1 ∼ 2 S2 if and only if λ(1 S1) = λ(2 S2). In particular, [S]=[λ(S)]. A2. The map ψ : C∗(B, F) → C,deﬁnedby[S] → λ(S), is a semigroup isomorphism. × ∗ C × ( , ) ∼ C ( , ) = C( , ) A3. F B F = F and B F B F . ∗ A4. CB (B, F) = E C (B, F) . Suppose these four statements hold true. Since B× ={1}=F× ∩ B, A2 implies that B ⊂ F is a C-monoid. Clearly, B ⊂ F is dense and by Theorem 1.1 B is seminormal. Since every class of C∗(B, F) contains a prime from C, all assumptions of Theorem 4.2 are satisﬁed and we obtain that H is half-factorial. 123 A. Geroldinger, Q. Zhong

Proof of A1. By definition of B, λ(1 S1) = λ(2 S2) implies that 1 S1 ∼ 2 S2. Con- versely, suppose 1 S1 ∼ 2 S2 and λ(1 S1) = λ(2 S2). If there exists an i ∈[0, n] such that λ(1 S1), λ(2 S2) ∈ Ci ,thena =−λ(1 S1) ∈ C ⊂ F and hence 1 S1a ∈ B, 2 S2a ∈/ B,a contradiction. After renumbering if necessary we may assume that λ(1 S1) ∈ Ci , λ(2 S2) ∈ C j for some i, j ∈[0, n] with i < j.Leta =−λ(1 S1) ∈ C ⊂ F.SinceGi G j and Ci + e j = C j , there exists gi ∈ Ci \{ei } such that gi + e j = e j .Then1 S1a ∈ B which implies that 2 S2a ∈ B. Thus 2 S2agi ∈ B which implies that 1 S1agi ∈ B.But λ(1 S1agi ) = gi = ei , a contradiction. To verify the in particular statement, note that λ(S) = ι() + σ(S) ∈ C ⊂ F and for an element g ∈ C we have λ(g) = g. Thus the claim follows immediately from the main statement. Proof of A2.ByA1, ψ is a well-defined monomorphism and obviously ψ is surjective. × × Proof of A3.Sinceι: F → C0 is a monomorphism, we have, using (2.2), that F = × × ∗ / ∼ C × ( , ) ⊂ C( , ) = C ( , ) F B = F B F C0 which implies that B F B F . ∗ Proof of A4. Theorem 1.1 implies that CB (B, F) ⊂ E C (B, F) .Conversely,letS ∈ F such that [S]∈E C∗(B, F) .ThenA2 implies that ι() + σ(S) ∈ E(C). Thus, by the definition of B, it follows that S ∈ B whence [S]∈CB (B, F).

Acknowledgements Open access funding provided by Austrian Science Fund (FWF). We thank the referee for their careful reading and all their comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and repro- duction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

References

1. Baginski, P., Chapman, S.T.: Congruence, arithmetic, monoids: a survey, combinatorial and additive number theory: CANT. 2012, Proceedings in Mathematics and Statistics, vol. 101, pp. 15–38. Springer, Berlin (2011) 2. Barucci, V.: Seminormal Mori domains, Commutative Ring Theory. Lect. Notes Pure Appl. Math., vol. 153, pp. 1–12. Marcel Dekker, New York City (1994) 3. Chapman, S.T., Coykendall, J.: Half-Factorial Domains, A Survey, Non-Noetherian Commutative Ring Theory, Mathematics and Its Applications, vol. 520, pp. 97–115. Kluwer Academic, Alphen aan den Rijn (2000) 4. Coykendall, J.: Extensions of half-factorial domains : a survey, Arithmetical Properties of Commutative Math Rings and Monoids, Lect. Notes Pure Appl, vol. 241, pp. 46–70. Chapman & Hall/CRC, Oxford (2005) 5. Coykendall, J., Malcolmson, P., Okoh, F.: Inert primes and factorization in extensions of quadratic orders. Houston J. Math. 43, 61–77 (2017) 6. Cziszter, K., Domokos, M., Geroldinger, A.: The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics, Multiplicative Ideal Theory and Factorization Theory, pp. 43–95. Springer, Berlin (2016) 7. Dobbs, D.E., Fontana, M.: Seminormal rings generated by algebraic integers. Mathematika 34, 141–154 (1987) 8. Foroutan, A., Hassler, W.: Factorization of powers in C-monoids. J. Algebra 304, 755–781 (2006) 9. Geroldinger, A., Halter-Koch, F.: Congruence monoids. Acta Arith. 112, 263–296 (2004) 10. Geroldinger, A., Halter-Koch, F.: Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol. 278. Chapman & Hall/CRC, Oxford (2006) 11. Geroldinger, A., Kainrath, F., Reinhart, A.: Arithmetic of seminormal weakly Krull monoids and domains. J. Algebra 444, 201–245 (2015) 12. Geroldinger, A., Ramacher, S., Reinhart, A.: On v-Marot Mori rings and C-rings. J. Korean Math. Soc. 52, 1–21 (2015) 123 A characterization of seminormal C-monoids

13. Geroldinger, A., Zhong, Q.: The set of distances in seminormal weakly Krull monoids. J. Pure Appl. Algebra 220, 3713–3732 (2016) 14. Gilmer, R.: Some questions for further research, Multiplicative Ideal Theory in Commutative Algebra, pp. 405–415. Springer, New York (2006) 15. Grillet, P.A.: Commutative Semigroups. Kluwer Academic, Alphen aan den Rijn (2001) 16. Halter-Koch, F.: Clifford semigroups of ideals in monoids and domains. Forum Math. 21, 1001–1020 (2009) 17. Halter-Koch, F., Hassler, W., Kainrath, F.: Remarks on the Multiplicative Structure of Certain One- Dimensional Integral Domains, Rings, Modules, Algebras, and Abelian Groups, Lect. Notes Pure Appl. Math., vol. 236, pp. 321–331. Marcel Dekker, New York City (2004) 18. Kainrath, F.: Arithmetic of Mori Domains and Monoids: The Global Case, Multiplicative Ideal Theory and Factorization Theory, Springer Proc. Math. Stat., vol. 170, pp. 183–218. Springer, Berlin (2016) 19. Malcolmson, P., Okoh, F.: Half-factorial subrings of factorial domains. J. Pure Appl. Algebra 220, 877– 891 (2016) 20. Oh, J.: On the algebraic and arithmetic structure of the monoid of product-one sequences. J. Commut. Algebra (to appear). https://projecteuclid.org/euclid.jca/1523433705 21. Oh, J.: On the algebraic and arithmetic structure of the monoid of product-one sequences II, Periodica Math. Hungarica (to appear). arxiv:1802.02851 22. Philipp, A.: A precise result on the arithmetic of non-principal orders in algebraic number ﬁelds. J. Algebra Appl. 11, 1250087 (2012). 42pp 23. Reinhart, A.: On integral domains that are C-monoids. Houston J. Math. 39, 1095–1116 (2013) 24. Schmid, W.A.: Half-factorial sets in ﬁnite abelian groups: a survey. Grazer Math. Ber. 348, 41–64 (2005) 25. Schmid, W.A.: Some Recent Results and Open Problems on Sets of Lengths of Krull Monoids with Finite Class Group, Multiplicative Ideal Theory and Factorization Theory, Springer Proc. Math. Stat., vol. 170, pp. 323–352. Springer, Berlin (2016)

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.

123