Numerical semigroups whose fractions are of maximal ... - Springer Link

Semigroup Forum (2011) 82: 412–422 DOI 10.1007/s00233-010-9275-5 R E S E A R C H A RT I C L E

Numerical semigroups whose fractions are of maximal embedding dimension David E. Dobbs · Harold J. Smith

Received: 25 June 2010 / Accepted: 29 October 2010 / Published online: 10 November 2010 © Springer Science+Business Media, LLC 2010

Abstract Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions Sk is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2 ≤ k ∈ N, S = Tk for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T . Let A (resp., F ) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain A = C1 ⊂ C2 ⊂ C3 ⊂ · · · ⊂ F , where, like A and F , each Cn is stable under the formation of fractions. Keywords Numerical semigroup · Maximal embedding dimension · Arf numerical semigroup · Saturated numerical semigroup · Multiplicity · Frobenius number · Fractionally closed

1 Introduction Let N denote the set of nonnegative integers. Let S be a numerical semigroup, i.e., an additive submonoid of N such that N \ S is finite. Convenient references on numerical semigroups include [1, 3, 4, 10]. Beginning with [12] and [8], several recent papers have considered the following notion of a quotient (also known as a fraction)

Communicated by Jorge Almeida. D.E. Dobbs () Department of Mathematics, University of Tennessee, Knoxville, TN 37996-0614, USA e-mail: [email protected] H.J. Smith Department of Mathematics and Physics, Thomas More College, Crestview Hills, KY 41017, USA e-mail: [email protected]

Numerical semigroups whose fractions are of maximal embedding

413

of a numerical semigroup. If S is a numerical semigroup and k is a positive integer, the quotient of S by k, denoted Sk , is defined as {n ∈ N | nk ∈ S}. It was noted in [12] that if S is a numerical semigroup and k is a positive integer, then Sk is a numerical semigroup that contains S. There has been considerable interest in determining which numerical semigroups can be expressed in the form Sk for suitable S drawn from historically important classes of semigroups. For instance, [5, 8, 9, 15] considered this question for S drawn from the class of symmetric semigroups; [5, 6, 15] for the class of pseudo-symmetric numerical semigroups (in the sense of [1, 3]); and [14] for the class of numerical semigroups of maximal embedding dimension. Related work also appeared in papers such as [13]. Our interest here is in studying the numerical semigroups S such that all the fractions Sk lie in certain classes of numerical semigroups, especially, the classes of saturated, Arf, and maximal embedding dimension. (The relevant definitions will be recalled below. For the moment, we note that saturated ⇒ Arf ⇒ maximal embedding dimension.) Since S1 = S, the numerical semigroups S in question must belong to the class being studied. In fact, that is the whole story for the saturated case and the Arf case, as Proposition 2.2 (resp., Proposition 2.3) shows that if the numerical semigroup S is saturated (resp., Arf), then so is Sk for each positive integer k. However, the story is more complicated for the class of numerical semigroups of maximal embedding dimension. On the one hand, it was shown in [14, Theorem 4.12] that if T is any numerical semigroup, then there exists a positive integer N such that if k ≥ N in N, then T = Sk for some S of maximal embedding dimension. However, Example 2.4 gives an example of a numerical semigroup of maximal embedding dimension with a fraction which is not of maximal embedding dimension. In a positive direction, Proposition 2.5 does give a sufficient condition for a numerical semigroup of maximal embedding dimension to be such that certain of its fractions are also of maximal embedding dimension, and Corollary 2.8 shows how the class of saturated (resp., of Arf; of maximal embedding dimension) numerical semigroups can be characterized in terms of fractions. The above naturally raises the question of studying classes C of numerical semigroups that are fractionally closed, in the sense that Sk ∈ C for all S ∈ C and all positive integers k. By the above comments, the classes of saturated semigroups and of Arf semigroups are each fractionally closed, but the class of semigroups of maximal embedding dimension is not fractionally closed. Located between the class A of Arf semigroups and the class E of semigroups of maximal embedding dimension is the class F of semigroups that are fractionally of maximal embedding dimension, in the sense that a numerical semigroup S ∈ F if and only if Sk is of maximal embedding dimension for all positive integers k. Not only are the containments A ⊂ F ⊂ E proper, but we construct in Example 2.14 and Corollary 2.15 an infinite strictly ascending chain A ⊂ C1 ⊂ C2 ⊂ · · · ⊂ F , where, like A and F , each Cn is also a fractionally closed class. The rest of this Introduction provides background and notational conventions that will be used below. Let S be a numerical semigroup. The notation S ∗ will be used to denote the set of nonzero elements of S. It is well known that every numerical semigroup is finitely generated and, in fact, has a unique minimal generating set. The embedding dimension of S, denoted e(S), is the cardinality of the unique minimal generating set of S. The multiplicity of S, denoted μ(S), is the smallest nonzero element of S. The largest element of Z\S is called the Frobenius number of S and

414

D.E. Dobbs, H.J. Smith

is denoted F (S); by definition, F (N) = −1. It is well known that e(S) ≤ μ(S). If e(S) = μ(S), we say that S is of maximal embedding dimension or is an MED numerical semigroup. For additional characterizations of MED numerical semigroups, see [1, Proposition I.2.9]; cf. also [11]. A numerical semigroup S is said to be saturated if, whenever s, s1 , . . . , sn ∈ S such that s ≥ si for each i ∈ {1, . . . , n} and z1 , . . . , zn ∈ Z such that z1 s1 + · · · + zn sn ≥ 0, then s + z1 s1 + · · · + zn sn ∈ S. For the purposes of this paper, it is best to say that a numerical semigroup S is Arf if, for all a, b, c ∈ S ∗ such that a ≥ b ≥ c, we have that a + b − c ∈ S. In [1, Theorem I.3.4], 15 other equivalent characterizations of Arf numerical semigroups are given. In Proposition 2.1, we prove that the above definition of “Arf” agrees with the usage in [1]. In any case, it is clear from the above definitions that saturated ⇒ Arf (use s = a, s1 = b, s2 = c, z1 = 1 and z2 = −1). Also, it is clear from conditions (iii) and (iv) of [1, Theorem I.3.4] that Arf ⇒ maximal embedding dimension. The interested reader is encouraged to show this directly using the above definition of “Arf”.

2 Results In this paragraph, we collect some background material that will be helpful, especially in the proofs of Propositions 2.1 and 2.5. To facilitate matters, S will always denote a numerical semigroup. As usual, we let n(S) := |{0, 1, . . . , F (S)} ∩ S|. If n := n(S), we indicate the elements of S with the notation S = {0 = s0 , s1 , . . . , sn−1 , sn , →}, where “→” means that every positive integer greater than sn lies in S. As in [3] (see also [1, pp. 1–4]), a relative ideal of S is a nonempty subset H of Z such that H + S ⊆ H and H + d ⊆ S for some d ∈ S. If H and K are relative ideals of S, then (H − K) := {z ∈ Z | z + K ⊆ H } is an example of a relative ideal of S. An ideal of S is a relative ideal of S which is contained in S. If i is a positive integer, then Si := {s ∈ S | s ≥ si } is an example of an ideal of S. Finally, if S is a numerical semigroup, H a proper ideal of S, and h the least (positive) integer in H , then (as in [1]) H is said to be stable if (H − H ) = H − h. The definition of “Arf numerical semigroup” given in the Introduction was taken from [13, p. 4]. It is known that this definition is equivalent to the 15 characterizations of “Arf” given in [1, Theorem I.3.4]. We next offer a short and easy proof. Proposition 2.1 Let S be a numerical semigroup. Then the following conditions are equivalent: (1) Si is a stable ideal for all 0 < i ≤ n(S); (2) S is Arf. Proof Let n := n(S) and write S = {0 = s0 , s1 , . . . , sn−1 , sn , →} as above. (1) ⇒ (2): Let a ≥ b ≥ c ∈ S ∗ . If c ≥ sn , then a + b − c ≥ sn and so a + b − c ∈ S. Therefore, we may suppose that c = si for some 0 < i < n. By (1), (Si −Si ) = Si −si . Since b ∈ Si , taking c := si gives that b − c ∈ Si − si = (Si − Si ). Hence, since a ∈ Si , it follows that a + (b − c) ∈ Si ⊆ S.


415

(2) ⇒ (1): We must show that (Si − Si ) = Si − si . To see that (Si − Si ) ⊆ Si − si , note that if x ∈ (Si − Si ), then x + Si ⊆ Si , whence x + si ∈ Si and x ∈ Si − si . We now prove the reverse inclusion. Let y ∈ Si − si and z ∈ Si . It suffices to prove that y + z ∈ Si . As y ∈ Si − si , we have y = a − si for some a ∈ Si . Thus, y + z = a + z − si . Since a, z ∈ Si , we have that a, z ≥ si . Regardless of how a and z compare with respect to ≥, (2) gives a + z − si ∈ S. Moreover, a + z − si ∈ Si since a, z ≥ si implies a + z − si ≥ si . We begin our study of various classes of numerical semigroups by asking the following. If T is saturated, what can be said about the typical fraction, Tk ? Proposition 2.2 gives the (easy) answer. Proposition 2.2 If T is a saturated numerical semigroup and k is any positive integer, then Tk is saturated. Proof Suppose T is saturated. Let S := Tk . Let s, s1 , . . . , sn ∈ S such that s ≥ si for each i ∈ {1, . . . , n} and z1 , . . . , zn ∈ Z such that z1 s1 + · · · + zn sn ≥ 0. We will show that s + z1 s1 + · · · + zn sn ∈ S. Note that ks, ks1 , . . . , ksn ∈ T since S = Tk . Moreover, ks ≥ ksi for each i ∈ {1, . . . , n} and kz1 s1 + · · · + kzn sn ≥ 0. Therefore, since T is saturated, ks + kz1 s1 + · · · + kzn sn ∈ T . Since ks + kz1 s1 + · · · + kzn sn = k(s + z1 s1 + · · · + zn sn ) and S = Tk , it follows that s + z1 s1 + · · · + zn sn ∈ S. Next, we answer the analogous question for the Arf case (namely, if T is Arf, what can be said about Tk ). Proposition 2.3 If T is an Arf numerical semigroup and k is any positive integer, then Tk is Arf. Proof Suppose T is Arf. Let S := Tk . Let a ≥ b ≥ c > 0 in S. Then ka ≥ kb ≥ kc > 0 in T . Since T is Arf, ka + kb − kc ∈ T . Note ka + kb − kc = k(a + b − c). Hence, a + b − c ∈ Tk = S. Thus, S is Arf. We next show that the above two results do not extend to the context of MED numerical semigroups. Example 2.4 If T := 3, 8, 13 = {0, 3, 6, 8, 9, 11, →}, then T is MED since e(T ) = 3 = μ(T ). However, T2 = {0, 3, 4, 6, →} = 3, 4 is not MED since e( T2 ) = 2 < 3 = μ( T2 ). Despite Example 2.4, we next get a result for “MED” in the spirit of Propositions 2.2 and 2.3 by using an additional hypothesis. The proof of Proposition 2.5 uses the following criterion for a numerical semigroup to be MED (see condition (vi) in [1, Theorem I.2.9]): a numerical semigroup S is MED if and only if the ideal S ∗ := S\{0} is stable. As in the proof of Proposition 2.1, (S ∗ − S ∗ ) ⊆ S ∗ − μ(S), and so this criterion is equivalent to S ∗ − μ(S) ⊆ (S ∗ − S ∗ ); i.e., that if a, b ∈ S ∗ , then a + b − μ(S) ∈ S.

416


Proposition 2.5 If T is MED and k is any positive integer such that k|μ(T ), then is MED.

T k

Proof Let S := Tk and let m := μ(T )/k. For any nonnegative integer x, we have x ∈ S if and only if kx ∈ T . Hence, μ(S) = min(S ∗ ) = min{x ∈ N | 0 < kx ∈ T } = m. Let a, b ∈ S ∗ . Since S = Tk , we have ka, kb ∈ T ∗ . Since T is MED, the above criterion gives ka +kb −μ(T ) ∈ T . Note that ka +kb −μ(T ) = ka +kb −km = k(a +b −m). Then a + b − m ∈ S since S = Tk , and so S is MED by the above criterion. Remark 2.6 (a) Example 2.4 shows that the hypothesis “k|μ(T )” in Proposition 2.5 cannot be weakened to “k ≤ μ(T )”. (b) The converse of Proposition 2.5 is not valid. Indeed, it is possible for S = Tk to be MED, yet T may not be MED or μ(T ) = kμ(S). For a first example of this, with k = 2, take S := 2, 3 = {0, 2, →} and T := 4, 5, 6 = {0, 4, 5, 6, 8, →}. Note that S is MED, S = T2 and 2μ(S) = μ(T ), but T is not MED. For a second example, take S := 2, 3 = {0, 2, →} and T := 3, 4, 5 = {0, 3, →}. In this example, S is MED, T is MED, and S = T2 , but 2μ(S) > μ(T ). Despite Remark 2.6(b), it is often possible to learn about a numerical semigroup from the behavior of one of its fractions. We next give a result along these lines. Proposition 2.7 Let S be a numerical semigroup, let 2 ≤ k ∈ N and let n ∈ S \ kN such that n > kF (S). Define Tn := kS ∪ {n + 1, →}. Then: (i) (ii) (iii) (iv)

Tn is a numerical semigroup and Tkn = S. If S is MED, then Tn is MED. If S is Arf, then Tn is Arf. If S is saturated, then Tn is saturated.

Proof It is straightforward to check the assertions if S = N, and so we assume henceforth that S = N. (i): Straightforward. For the remaining assertions, note that μ(Tn ) is either n + 1 or kμ(S). The case μ(Tn ) = n + 1 can arise (consider, for example, k := 3, n := 4 and S := 2, 3 ). However, any numerical semigroup of the form {0, n + 1, →} is easily seen to be (MED, Arf and) saturated. Thus, in proving (ii)–(iv), we may assume, without loss of generality, that μ(Tn ) = kμ(S). (ii) We show that if a, b ∈ Tn∗ , then a + b − μ(Tn ) ∈ Tn . If a ≥ n + 1, then a + b − μ(Tn ) ≥ a + μ(Tn ) − μ(Tn ) = a ≥ n + 1 > F (Tn ), so a + b − μ(Tn ) ∈ Tn . Hence, we may suppose that a, b < n + 1. Then a, b ∈ kS, with a = ksa and b = ksb for some sa , sb ∈ S ∗ . As S is MED, sa + sb − μ(S) ∈ S. Therefore, since S = Tkn , k(sa + sb − μ(S)) ∈ Tn . But k(sa + sb − μ(S)) = ksa + ksb − kμ(S) = a + b − μ(Tn ). (iii) We show that if a, b, c ∈ Tn∗ with a ≥ b ≥ c, then a + b − c ∈ Tn . If a ≥ n + 1, then a + b − c ≥ a + b − b = a ≥ n + 1 > F (Tn ), so a + b − c ∈ Tn . Hence, we may suppose that a < n + 1. Then a, b, c ∈ kS, with a = ksa , b = ksb and c = ksc for some sa , sb , sc ∈ S ∗ , and sa ≥ sb ≥ sc . As S is Arf, sa + sb − sc ∈ S. Therefore, since S = Tkn , k(sa + sb − sc ) ∈ Tn . But k(sa + sb − sc ) = ksa + ksb − ksc = a + b − c.


417

(iv) Let t, t1 , . . . , tr ∈ Tn such that t ≥ ti for each i ∈ {1, . . . , r} and z1 , . . . , zr ∈ Z such that z1 t1 + · · · + zr tr ≥ 0. We show that t + z1 t1 + · · · + zr tr ∈ Tn . If t ≥ n + 1, then t + z1 t1 + · · · + zr tr ≥ t ≥ n + 1 > F (Tn ), so the assertion holds. Hence, we may suppose that t < n + 1. Then t, t1 , . . . , tr ∈ kS, with t = ks and ti = ksi for each i ∈ {1, . . . , r}, for some s, s1 , . . . , sr ∈ S, and s ≥ si for each i ∈ {1, . . . , r}. Since z1 t1 + · · · + zr tr = z1 ks1 + · · · + zr ksr = k(z1 s1 + · · · + zr sr ) ≥ 0, we have z1 s1 + · · · + zr sr ≥ 0. As S is saturated, s + z1 s1 + · · · + zr sr ∈ S. Then k(s + z1 s1 + · · · + zr sr ) ∈ Tn since S = Tkn . But k(s + z1 s1 + · · · + zr sr ) = ks + z1 ks1 + · · · + zr ksr = t + z1 t1 + · · · + zr tr . By combining the above Propositions, we obtain the following characterization of the “MED”, “Arf” and “saturated” concepts in terms of fractions. Note that a harder proof of the assertion regarding the “MED” condition in Corollary 2.8 was given in [14, Theorem 3.6]. Corollary 2.8 Let S be a numerical semigroup. Then S is MED (resp., Arf; resp., saturated) if and only if, for each 2 ≤ k ∈ N, S = Tk for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T . Proof A proof of the first assertion (that is, characterizing S of MED), appears in [14, Theorem 3.6]. For an alternate proof of the “only if” part of this assertion (i.e., assuming S is MED), one can apply Proposition 2.7(i), since it is easy to see (using the notation of Proposition 2.7) that there are infinitely many distinct numerical semigroups of the form Tn . The assertions for “Arf” and “saturated” follow easily by combining Propositions 2.2, 2.3 and 2.7(iii), (iv) with the last comment in the preceding sentence. In general, it is quite difficult to answer the following type of question: what can be said about a numerical semigroup T if Tk is MED (resp., Arf; resp., saturated) for a specific k? It is possible that Tk is MED (resp., Arf; resp., saturated) while T is not. The following proposition shows that there is only one S and one k for which every T satisfying S = Tk is Arf (resp., saturated). Proposition 2.9 Let S be an Arf (resp., saturated; resp., MED) numerical semigroup, let k ≥ 2 be a positive integer, and let Tk := Tk,S := {numerical semigroups T | S = T k }. Then T is Arf (resp., saturated; resp., MED) for all T ∈ Tk if and only if S = N and k = 2. Proof (⇐) If T2 = N, then T has the form T = 2N ∪ {a, →} for some a ∈ N. It is straightforward to verify that any such T is saturated (hence, Arf and MED). (⇒) We prove the “saturated” and “Arf” assertions first. Suppose first that S = N. We will find T ∈ Tk such that T is not Arf (and hence not saturated). It is convenient to let c := F (S) + 1 (≥ 2). If k > 2, let T := kS ∪ {kc + k, kc + k + 1, kc + 2k, →}. It is easy to verify that T is a numerical semigroup and that Tk = S; i.e., T ∈ Tk . Note that kc + k + 1 ≥ kc + k + 1 ≥ kc + k and 2(kc + k + 1) − (kc + k) = kc + k + 2. As k > 2, kc + k + 2 ∈ / kS and kc + k + 1 < kc + k + 2
2 and S = N. We will find T ∈ Tk such that T is not MED (and hence neither saturated nor Arf). It suffices to find a numerical semigroup T which is not MED but satisfies μ(T ) = k. Therefore, taking T := k, k + 1 suffices. This completes the proof of the “saturated” and “Arf” assertions. We have only to prove that if k ≥ 2 and S = N but S is MED, then there exists T ∈ Tk such that T is not MED. Let {s1 , . . . , sm } be the minimal generating set of S, with s1 < · · · < sm . Choose n ∈ S such that n > ksm and gcd(n, ks1 , . . . , ksm ) = 1; i.e., gcd(n, k) = 1. Consider the numerical semigroup T := ks1 , . . . , ksm , n . Since k ≥ 2 and S = N, we have μ(T ) = ks1 ≥ s1 + s1 > s1 + 1 = e(T ), where the last step holds via the MED property of S since s1 + 1 = μ(S) + 1 = e(S) + 1 = m + 1 = e(T ), and so T is not MED. On the other hand, Tk = S. Indeed, it is clear that S ⊆ Tk . To see that Tk ⊆ S, consider any λ ∈ Tk . As kλ ∈ T = kS + nN, there exist s ∈ S and ν ∈ N such that kλ = ks + νn. Since gcd(n, k) = 1, we have that k|ν, say with ν = αk for some α ∈ N. Canceling k leads to λ = s + αn ∈ S + S = S, as desired. By Proposition 2.3, each Arf numerical semigroup is an MED numerical semigroup with the property that each of its fractions is MED. A natural question to ask is the following: is the class of Arf numerical semigroups maximal with respect to this property? To use notation from the Introduction, we are asking whether A = F . The answer is “No” and leads us to view F as a new class of numerical semigroups that incorporates, in some sense, a more natural generalization of the “MED” property than did the “Arf” property. Let T be a numerical semigroup. We say that T is fractionally MED (in short, FMED) if, for all positive integers k, the numerical semigroup Tk is MED. Note that all Arf or saturated numerical semigroups are FMED, by Propositions 2.2 and 2.3. It is easy to see that FMED ⇒ MED (just take k = 1), and so F ⊆ E (using more notation from the Introduction). Moreover, Example 2.4 shows that MED ⇒ FMED, and so F ⊂ E. We next show that “FMED” admits a characterization in the spirit of the definitions of “saturated” and “Arf” that were given in the Introduction. First, it is convenient to introduce the following. If T is a numerical semigroup and k is a positive integer, then the k-multiplicity of T is defined as μk (T ) := min(kN ∩ T ∗ ). Note that μ1 (T ) = μ(T ). Proposition 2.10 Let T be a numerical semigroup and k a positive integer. Then is MED if and only if a + b − μk (T ) ∈ T for all a ≥ b in kN ∩ T ∗ .

T k

Proof (⇒) Let S := Tk , and let a ≥ b in kN ∩ T ∗ . Since S = Tk , we have that a/k ≥ b/k in S ∗ . But S = Tk is MED by hypothesis, and so a/k + b/k − μ(S) ∈ S. Note that μ(S) = min(kN ∩ T ∗ )/k = μk (T )/k. Thus, a/k + b/k − μk (T )/k ∈ S. It follows that a + b − μk (T ) = k(a/k + b/k − μk (T )/k) ∈ kS ⊆ T .


419

(⇐) Let S := Tk . To prove that S is MED, it suffices to show that a + b − μ(S) ∈ S for any a ≥ b in S ∗ . Since S = Tk , we have that ka ≥ kb in T ∗ . By hypothesis, ka + kb − μk (T ) ∈ T . Moreover, since μ(S) = min(kN ∩ T ∗ )/k, we have that kμ(S) = μk (T ). Hence, ka + kb − kμ(S) ∈ T and, since S = Tk , it follows that a + b − μ(S) ∈ S. Corollary 2.11 A numerical semigroup T is FMED if and only if a + b − μk (T ) ∈ T for all positive integers k and for all a ≥ b in kN ∩ T ∗ . We next show that “FMED” is a genuinely new concept. Example 2.12 FMED ⇒ Arf. Indeed, T := 5, 9, 13, 16, 17 = {0, 5, 9, 10, 13, →} is FMED but not Arf. To prove that T is not Arf, observe that 10 + 10 − 9 = 11 ∈ / T . We show next that T is FMED. Let 2 ≤ k ∈ N. If k ∈ T , then Tk = N, which is MED. Therefore, it suffices to show that Tk is MED for all positive integers k ∈ {1, 2, 3, 4, 6, 7, 8, 11, 12}. Simple calculations show that T1 = T , T2 = 5, 7, 8, 9, 11 , T3 = 3, 5, 7 , T4 = 4, 5, 6, 7 , T T T T T 6 = 3, 4, 5 , and 7 = 8 = 11 = 12 = 2, 3 . Hence, T is FMED. The last six fractions that were considered in the proof of Example 2.12 were each of the form {0, n + 1, →}. It is interesting to note that any numerical semigroup having such a form must be MED. Remark 2.13 Let S be a numerical semigroup and let a ∈ S ∗ . If S is MED, then [1, Proposition I.2.9] shows that (S + a) ∪ {0} is a MED numerical semigroup. Moreover, if S is Arf, then [10, Proposition 3.18] shows that (S + a) ∪ {0} is Arf. It is interesting to note that FMED numerical semigroups do not exhibit similar behavior. For example, if S = 6, 13, 20, 27, 28, 35 then it can be shown that S is FMED, but (S + 6) ∪ {0} is not FMED (the quotient of (S + 6) ∪ {0} over 5 is the numerical semigroup 5, 6, 8, 9 , which is not MED). Example 2.14 will sharpen the impact of Example 2.12. Example 2.14 For each integer n ≥ 2, let Sn := {0, 2n , 2 · 2n , 9 · 2n−2 , 3 · 2n , →}. Then each such Sn is an FMED numerical semigroup which is not Arf. Moreover, for each integer n ≥ 2, Sn+1 ⊂ Sn . Hence, S2 ⊃ S3 ⊃ S4 ⊃ S5 ⊃ · · · is an infinite strictly descending chain of FMED numerical semigroups, none of which is Arf. For a proof, it is clear that each Sn is a numerical semigroup. Moreover, if m > n ≥ 2 in N, then Sm ⊂ Sn . Indeed, the containment is easy to check; and it is a proper inclusion since 2n ∈ Sn \ Sm . It remains only to show that each Sn is FMED but not Arf. To show that Sn is not Arf, note that a := b := 9 · 2n−2 and c := 2n+1 are each elements of Sn satisfying a ≥ b ≥ c, although a + b − c = 9 · 2n−2 + 9 · 2n−2 − 2n+1 = 10 · 2n−2 ∈ Sn

420


since 9 · 2n−2 < 10 · 2n−2 < 3 · 2n and 9 · 2n−2 + 2n > 8 · 2n−2 + 2n = 3 · 2n . Finally, we will show that each Sn is FMED, namely, that Skn is MED for each positive integer k. By the criterion in Proposition 2.10, we must show that a + b − μk (Sn ) ∈ Sn whenever a ≥ b > 0 in kN ∩ Sn . If μk (Sn ) ≥ 3 · 2n , this criterion is clearly satisfied since b ≥ μk (Sn ) and {3 · 2n , →} ⊆ Sn . Thus, we may assume, without loss of generality, that k is not divisible by a prime number other than 2 or 3. We consider first the case k = 1. As S1n = Sn ,we must show that a + b − 2n = a + b − μ(Sn ) ∈ Sn whenever a ≥ b > 0 in Sn . As we can ignore the trivial case b = 2n and {3 · 2n , →} ⊆ Sn , it suffices to observe that a + b − 2n ≥ 2n+1 + 2n+1 − 2n = 3 · 2n ∈ Sn . The preceding reasoning applies whenever μk (Sn ) = 2n . In particular, the criterion from Proposition 2.10 is satisfied if k = 2ν where ν is any positive integer such that ν ≤ n. As μk (Sn ) ≥ k in general and we have seen that the criterion is satisfied if μk (Sn ) ≥ 3 · 2n , it follows that the only remaining multiple of 2 that needs to be examined is k = 2n+1 . In this case, μk (Sn ) = 2n+1 , and so our task is to prove that if a ≥ b > 0 in 2n+1 N ∩ Sn , then a + b − 2n+1 ∈ Sn . Without loss of generality, b ≥ 2n+2 . Then a + b − 2n+1 ≥ 2n+2 + 2n+2 − 2n+1 = 3 · 2n+1 , and so a + b − 2n+1 ∈ Sn . The only remaining values of k that need to be checked are 3 and 9. For these values of k, note that μk (Sn ) = 9 · 2n−2 (since 9 · 2n−2 < 3 · 2n ). The only nontrivial thing to check is that if a ≥ b > 0 in 3N ∩ Sn , then a + b − 9 · 2n−2 ∈ Sn . Without loss of generality, b > 9 · 2n−2 . Then b ≥ 3 · 2n and the assertion is clear, for then a + b − 9 · 2n−2 > a ≥ 3 · 2n . This completes the proof. Recall from the Introduction that a class C of numerical semigroups is said to be fractionally closed if, for all positive integers k, Sk ∈ C whenever S ∈ C. We have seen that the classes of Arf and of saturated numerical semigroups are each fractionally closed. Moreover, by the “a over b” lemma [6, Lemma 11], the class F of fractionally MED numerical semigroups constitutes another such class and it is clearly the largest fractionally closed class within E. By Examples 2.4 and 2.12, A ⊂ F ⊂ E. It is natural to ask if there are other fractionally closed classes of numerical semigroups between A and F . We next use Example 2.14 to produce a chain of infinitely many other fractionally closed fractionally MED classes of numerical semigroups. Corollary 2.15 There exists an infinite strictly ascending chain A = C1 ⊂ C2 ⊂ C3 ⊂ · · · ⊂ F , where, like A and F , each Cn is fractionally closed (i.e., stable under the formation of fractions). Proof Put C1 := A. Consider the data from Example 2.14. For each integer n ≥ 2, let Sm for some positive integers m ≤ n and k . Cn := A ∪ T ∈ F | T = k


421

In other words, if n ≥ 2, then Cn is the union of A with the set of all fractions of the semigroups S2 , . . . , Sn . By Proposition 2.3, Example 2.14 and the “a over b” lemma, each Cn is a fractionally closed class of FMED numerical semigroups; in particular, Cn ⊆ F . Also, if n ≥ 2, then Cn ⊃ A since Sn is not Arf. It is clear that Cn ⊆ Cn+1 for each n. These inclusions are proper, since Sn+1 ∈ Cn+1 \ Cn if n ≥ 2. For otherwise, Sn+1 = Skm with 1 ≤ m ≤ n and Sn ⊃ Sn+1 =

Sm ⊇ Sm ⊇ Sn , k

a contradiction. Moreover, since Cn ⊂ Cn+1 ⊆ F , we also have Cn ⊂ F for all n.

Note that the data in Corollary 2.15 satisfy n Cn ⊂ F . Indeed, it can be shown that 4, 25, 30, 31 ∈ F \ n Cn . We close with a remark amplifying some comments of the referee. Remark 2.16 (a) The notion of an Arf numerical semigroup has been generalized in [2] to the notion of a numerical semigroup that admits a given strongly admissible pattern. The latter notion was defined in [2, p. 656], with the Arf case corresponding to the pattern p = x1 + x2 − x3 [2, Example 2]. The referee has kindly noted that our results on Arf numerical semigroups can be generalized to numerical semigroups that admit a given strongly admissible pattern. For the most part, we leave to the reader the task of verifying the associated details. However, we wish to point out the following. Proposition 2.3 can be generalized (with much the same proof as before) to state that if a numerical semigroup S admits a pattern p, then so does each fraction S k . In particular, the class S(p) of numerical semigroups admitting p is fractionally closed. (b) Recall from [7] that a Frobenius variety is defined as a nonempty family V of numerical semigroups which is closed under finite intersections such that S ∪ {g(S)} ∈ V for all S ∈ V with S = N. The classes of Arf and of saturated numerical semigroups each form a Frobenius variety, whereas the class of MED numerical semigroups is not a Frobenius variety (cf. [10, Chap. 6]). As we have seen, the classes of Arf and of saturated numerical semigroups are each fractionally closed, whereas the class of MED numerical semigroups is not fractionally closed. The referee has kindly asked whether this coincidence points to a possible connection between the notions of “Frobenius variety” and “fractionally closed”. In this regard, we can make the following observations. The book [10] includes the following four examples of classes of numerical semigroups that form a Frobenius variety: (a) Arf; (b) saturated; (c) numerical semigroups having a Toms decomposition; (d) numerical semigroups that admit a given pattern. We have been able to prove that each of these four classes is fractionally closed. (All these proofs are short and simple.) However, we have also been able to construct a Frobenius variety that is not fractionally closed; and a fractionally closed class of numerical semigroups which is not a Frobenius variety. For this reason, we conclude that the referee’s question can be answered thus: it is a coincidence!

422


References 1. Barucci, V., Dobbs, D.E., Fontana, M.: Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Mem. Am. Math. Soc. 125(598) (1997) 2. Bras-Amorós, M., García-Sánchez, P.A.: Patterns on numerical semigroups. Linear Algebra Appl. 414, 652–669 (2006) 3. Fröberg, R., Gottlieb, C., Häggkvist, R.: On numerical semigroups. Semigroup Forum 35, 63–83 (1987) 4. Ramírez Alfonsín, J.L.: The Diophantine Frobenius Problem. Oxford Lecture Series Math. and Its Appl., vol. 30. Oxford Univ. Press, Oxford (2005) 5. Robles-Pérez, A.M., Rosales, J.C., Vasco, P.: The doubles of a numerical semigroup. J. Pure Appl. Algebra 213, 387–396 (2009) 6. Rosales, J.C.: One half of a pseudo-symmetric numerical semigroup. Bull. Lond. Math. Soc. 40, 347– 352 (2008) 7. Rosales, J.C.: Families of numerical semigroups closed under finite intersections and for the Frobenius number. Houst. J. Math. 34, 339–348 (2008) 8. Rosales, J.C., García-Sánchez, P.A.: Every numerical semigroup is one half of a symmetric numerical semigroup. Proc. Am. Math. Soc. 136, 475–477 (2008) 9. Rosales, J.C., García-Sánchez, P.A.: Every numerical semigroup is one half of infinitely many symmetric numerical semigroups. Commun. Algebra 36, 2910–2916 (2008) 10. Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups. Developments in Math., vol. 20. Springer, New York (2009) 11. Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Branco, M.B.: Numerical semigroups with maximal embedding dimension. Int. J. Commut. Rings 2, 47–53 (2003) 12. Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Urbano-Blanco, J.M.: Proportionally modular diuophantine inequalities. J. Number Theory 103, 281–294 (2003) 13. Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Branco, M.B.: Arf numerical semigroups. J. Algebra 276, 3–12 (2004) 14. Smith, H.J.: Numerical semigroups that are fractions of numerical semigroups of maximal embedding dimension. JP J. Algebra Number Theory Appl. 17, 69–96 (2010) 15. Swanson, I.: Every numerical semigroup is one over d of infinitely many symmetric numerical semigroups. In: Commutative Algebra and its Applications, pp. 383–386. de Gruyter, Berlin (2009)