The Oversemigroups of a Numerical Semigroup - Universidad de

Download PDF
0 downloads 0 Views 152KB Size Report
Comment
S∈V. S is bijective, with the inverse bijection given by M ↦→ {S ∈ S | M ⊆ S}. Moreover V is a finite variety if and only if P(V) is a numerical semigroup. Proof.
Semigroup Forum Vol. 67 (2003) 145–158

c 2003 Springer-Verlag New York Inc. 

DOI: 10.1007/s00233-002-0007-3

RESEARCH ARTICLE

The Oversemigroups of a Numerical Semigroup J. C. Rosales, P. A. Garc´ıa-S´ anchez, J. I. Garc´ıa-Garc´ıa, and J. A. Jim´ enez Madrid Communicated by Jimmie D. Lawson

Abstract We study the set of numerical semigroups containing a given numerical semigroup. As an application we prove characterizations of irreducible numerical semigroups that unify some of the existing characterizations for symmetric and pseudo-symmetric numerical semigroups. Finally we describe an algorithm for computing a minimal decomposition of a numerical semigroup in terms of irreducible numerical semigroups.

Introduction Let N be the set of nonnegative integers. A subset M of N is a submonoid of N if it is closed under addition and 0 ∈ H . Given A ⊆ N , define A =

 k 

 ni ai : k ∈ N, a1 , . . . , ak ∈ A, n1 , . . . , nk ∈ N .

i=1

Clearly A is a submonoid of N , which is known as the submonoid of N generated by A . For a submonoid M of N , we say that A ⊆ M is a system of generators of M if M = A. If no proper subset of such A generates M , then A is a minimal system of generators of M . A numerical semigroup S is a submonoid of N that generates Z as a group (Z denotes the set of integers). From this definition the following results can be easily deduced (see for instance [2] or [6]). • The set N\S is finite. The largest integer not belonging to S is known as the Frobenius number of S and it is denoted by g(S) . • The numerical semigroup S has a minimal system of generators {n1 < · · · < np } = (S\{0})\((S\{0}) + (S\{0})) . The integer n1 , known as the multiplicity of S and denoted by m(S) , is the least positive integer in S , and p is the embedding dimension of S , denoted by e(S) .

146

Rosales et al.

A numerical semigroup is irreducible if it cannot be expressed as an intersection of two numerical semigroups containing it properly. In [11] it is shown that using the results appearing in [2], [3], one can prove that the class of irreducible numerical semigroups with odd (even) Frobenius number coincides with the class of symmetric (pseudo-symmetric) numerical semigroups. Kunz shows in [4] that a numerical semigroup is symmetric if and only if its associated semigroup ring k[[S]] is Gorenstein, and in [2] it is demonstrated that the same correspondence holds for pseudo-symmetric numerical semigroups and Kunz rings. Every numerical semigroup can be expressed as an intersection of finitely many irreducible numerical semigroups. The study of such a decomposition has been recently treated in [9], [8], [10], [11]. A natural question concerns finding a decomposition with the least number of irreducibles involved. In this paper we give an algorithmic method to construct a minimal decomposition of any numerical semigroup as an intersection of irreducible numerical semigroups. The achievement of this procedure led us to the study of the set of all numerical semigroups containing a fixed numerical semigroup; thus the title of the paper. The contents of this work are organized as follows. In Section 1 we introduce the concept of variety of numerical semigroups as a family of numerical semigroups closed under extensions and finite intersections. Theorem 4 gives several characterizations of this concept, and we show that a variety is the set of all numerical semigroups containing a given submonoid of N . Along this line, Corollary 5 provides a one-to-one correspondence between varieties and submonoids of N . This correspondence maps finite varieties to numerical semigroups; whence a finite variety is just the set of all numerical semigroups containing a fixed numerical semigroup. Section 2 is devoted to the study of finitely generated varieties. Proposition 8 states that the concepts of finitely generated variety, cyclic variety and finite variety are all the same. In Section 3 for a given numerical semigroup S we define the set EH(S) as the set of positive integers x not belonging to S such that S ∪ {x} is a numerical semigroup. The properties related to this set allows us to construct algorithmically the (finite) variety of all numerical semigroups containing S . Section 4 focuses on the study of the set EH(S) , and there we show that S is a maximal numerical semigroup with respect to the constraint S ∩ EH(S) = ∅ . We also prove that if A ⊆ N and S is maximal not cutting A , then EH(S) ⊆ A . Thus EH(S) is the least subset of N\S that determines S up to maximality. Finally in Section 5 we use all these results to give an algorithm for computing the minimal decomposition in irreducibles of a given numerical semigroup (minimal with respect to the number of irreducibles used in the decomposition). 1. Varieties of numerical semigroups A nonempty family V of numerical semigroups is a variety if the following holds: (1) if S ∈ V and S ⊆ S  , then S  ∈ V , (2) if S, S  ∈ V , then S ∩ S  ∈ V .

Rosales et al.

147

Given a family of numerical semigroups F , denote by P(F) =



S

S∈F

(if F is empty, just set P(F) = N ). Observe that P(F) is a submonoid of N , which in general does not have to be a numerical semigroup. However, if F has finitely many elements, then P(F) is a numerical semigroup (since the intersection is clearly a monoid, and its complement in N is still finite; note that the Frobenius number of the resulting semigroup is the maximum of the Frobenius numbers of the semigroups involved in the intersection). It is easy to check (see [5]) that if S is a numerical semigroup, S = N , then S ∪ {g(S)} is again a numerical semigroup. Thus we obtain the following straightforward result. Lemma 1.

Let V be a variety. If S ∈ V , then S ∪ {g(S)} ∈ V .

For a numerical semigroup S , we can recurrently define Sn as follows: • S0 = S , • Sn+1 =



Sn ∪ {g(Sn )}, N,

if Sn = N, otherwise.

If k = #(N\S) (#A stands for cardinality of A ), then it is clear that S = S0  S1  · · ·  Sk = N. Given F a family of numerical semigroups, define g(F) = {g(S): S ∈ F}. Lemma 2. Let F be a family of numerical semigroups such that for every S ∈ F , S = N , we have that S ∪ {g(S)} ∈ F . Then P(F) = N\g(F). Proof. Let x ∈ P(F) . Then x ∈ S for all S ∈ F , whence x = g(S) for all S ∈ F . It follows that x ∈ N\g(F) . Assume that there exists x ∈ N\g(F) such that x ∈ P(F) . Then there is S ∈ F with x ∈ S . Using the sequence S0  · · ·  Sk given above, there exists n such that g(Sn ) = x. By hypothesis Sn ∈ F , whence x ∈ g(F) , a contradiction. Lemma 3. Let V be a variety and let S be a numerical semigroup such that S ∪ {g(S)} ∈ V and g(S) ∈ g(V). Then S ∈ V . Proof. As g(S) ∈ g(V) , there exists S  ∈ V with g(S  ) = g(S) . Since  S ∩ S = (S ∪ {g(S)}) ∩ S  ∈ V , we have that S ∈ V (since S ∩ S  ⊆ S ).

148

Rosales et al. Denote by S the set of all numerical semigroups.

Theorem 4. Let F be a nonempty family of numerical semigroups. The following statements are equivalent. (1) F is a variety. (2) F fulfills the following conditions: (i) if S ∈ F , S = N , then S ∪ {g(S)} ∈ F , (ii) for every S ∈ S , if S ∪ {g(S)} ∈ F and g(S) ∈ g(F), then S ∈ F . (3) F fulfills the following conditions: (i) N ∈ F , (ii) for any numerical semigroup S , if S ∪{g(S)} ∈ F and g(S) ∈ P(F), then S ∈ F . (4) F = {S ∈ S: P(F) ⊆ S} . (5) There exists a subset A of N such that F = {S ∈ S: A ⊆ S} . (6) There exists a finite subset A of N for which F = {S ∈ S: A ⊆ S}. Proof. (1) implies (2) follows from Lemmas 1 and 3. (2) implies (3) is a consequence of the sequence S = S0  · · ·  Sk = N given just before Lemma 2 and of Lemma 2 itself. For proving (3) implies (4), it suffices to show that if S ∈ S and P(F) ⊆ S , then S ∈ F . We use induction on #(N\S) . If #(N\S) = 0 , then S = N ∈ F . Assume by induction hypothesis that if S  ∈ S is such that P(F) ⊆ S  and #(N\S  ) = n , then S  ∈ F . Take now S ∈ S with #(N\S) = n + 1 and P(F) ⊆ S , and let S¯ = S ∪ {g(S)} . Using the induction hypothesis ¯ = n and P(F) ⊆ S¯ ), we have that S¯ is in F . Since g(S) ∈ S and ( #(N\S) P(F) ⊆ S , we have that g(S) ∈ P(F) , whence S ∈ F . (4) implies (5) is trivial. For (5) implies (6), just note that for any subset A of N , one gets that {S ∈ S: A ⊆ S} = {S ∈ S: A ⊆ S}. Since every submonoid of N is finitely generated, we deduce that there exists {a1 , . . . , at } ⊆ N such that a1 , . . . , at  = A, whence {S ∈ S: A ⊆ S} = {S ∈ S: {a1 , . . . , at } ⊆ S}. (6) implies (1) follows directly from the definition of variety.

Rosales et al.

149

The map

Corollary 5.

P: {V : V is a variety} → {M : M is a submonoid of N},  P(V) = S∈V S is bijective, with the inverse bijection given by M → {S ∈ S | M ⊆ S}. Moreover V is a finite variety if and only if P(V) is a numerical semigroup. Proof. The map is into due to (4) in Theorem 4. Let M be a submonoid of N and let V = {S ∈ S: M ⊆ S} , which by (5) in Theorem 4 is a variety. If we prove that P(V) = M , then we have that P is onto. Clearly M ⊆ P(V) . Now take x ∈ N\M . Then x does not belong to the numerical semigroup S = M ∪ {x + 1, →} (the symbol → is used here to indicate that all subsequent elements are in the set). Since S ∈ V and x ∈ S , we have that x ∈ P(V) . This proves that P(V) ⊆ M . If V is finite, then P(V) is a numerical semigroup, since the intersection of finitely many numerical semigroups is again a numerical semigroup. For the converse, assume that P(V) is a numerical semigroup. Then N\P(V) has finitely many elements and thus V = {S ∈ S: P(V) ⊆ S} can only have finitely many elements. 2. Finitely generated varieties For A ⊆ N , set

V(A) = {S ∈ S: A ⊆ S}.

In view of Theorem 4, V(A) is a variety. If we are given {Ai }i∈I a family of subsets of N , then     V(Ai ) = V Ai . i∈I

i∈I

Thus we have the following result. Lemma 6.

The intersection of varieties is a variety.

This result allows us to introduce the concept of variety generated by a family of numerical semigroups. Let F ⊆ S . We denote by F the intersection of all varieties containing F . By Lemma 6, F is a variety, and it is the smallest (with respect to set inclusion) variety containing F . We say that F is a system of generators of F. A variety V is finitely generated if it has a finite system of generators. As a consequence of Theorem 4 we obtain the following. Lemma 7.

Let F be a family of numerical semigroups. Then F = {S ∈ S: P(F) ⊆ S}.

We say that a variety V is cyclic if there exists a numerical semigroup S such that V = {S}.

150

Rosales et al. Let V a variety. The following conditions are equivalent.

Proposition 8. (1) V is cyclic.

(2) V is finitely generated. (3) V is finite. Proof. (1) implies (2) is trivial. If F is a finite system of generators of V , then P(F) is a numerical semigroup, as each is the minimal element of V . In view of Theorem 4 and Lemma 7, we have that P(V) = P(F) . Corollary 5 states that V is finite. This proves (2) implies (3). Now assume that V is finite. Then P(V) is a numerical semigroup. Lemma 7 together with Theorem 4 ensure that V = {P(V)}. 3. Unitary extensions of a numerical semigroup Given a numerical semigroup S , denote by H(S) = N\S,

EH(S) = {x ∈ H(S): 2x ∈ S, x + s ∈ S for all s ∈ S\{0}}.

Using this definition, it is straightforward to prove the next proposition, which describes those elements that added to a numerical semigroup yield a numerical semigroup. Proposition 9. Let S be a numerical semigroup and let x ∈ H(S). The following properties are equivalent: (1) x ∈ EH(S), (2) S ∪ {x} is a numerical semigroup. Every numerical semigroup containing properly the numerical semigroup S must contain an element of EH(S) as the following straightforward result shows. Lemma 10. Let S and S¯ be two numerical semigroups such that S  S¯ . ¯ Then S ∪ {max(S\S)} is a numerical semigroup. For two numerical semigroups S  S¯ , if we replace N in the sequence S0 = S  S1 = S0 ∪ {g(S)}  · · ·  N described in Section 1 with S¯ , then we ¯ i ) , obtaining the sequence have to replace g(Si ) with max(S\S • S0 = S , • Sn+1 =



Sn , ¯ n )}, Sn ∪ {max(S\S

¯ if Sn = S, otherwise.

Rosales et al.

151

¯ And if #(S\S) = k , then ¯ S = S0  · · ·  Sk = S. In view of Lemma 10, Si is a numerical semigroup for all i. With this remark one can actually compute all numerical semigroups containing a numerical semigroup S , that is, the finite variety V(S) . The idea is to proceed as follows: once you have an element S  of V(S) (you start with S  = S ), we get that S  ∪ {x1 }, . . . , S  ∪ {xr } ∈ V(S) , with {x1 , . . . , xr } = EH(S  ) ; then do the same with each of the S  ∪ {xi } . Example 11. Let S = 5, 7, 9, 11. For this semigroup, EH(S) = {13} and thus for obtaining a semigroup adding only an element, we only can adjoin its Frobenius number ( 13 in this case). So we get the semigroup S ∪ {13} . This semigroup is equal to 5, 7, 9, 11, 13 and EH(S ∪ {13}) = {6, 8} . So, from this one we obtain two new semigroups which are S ∪ {13, 6} and S ∪ {13, 8} . Performing this process as many times as necessary we get all the elements of V(S) . The variety V(S) is described in the following graph.

4. The set EH(S) As a consequence of Lemma 10, if S is a numerical semigroup, then we have that S is maximal in the set of all numerical semigroups not meeting EH(S) . Moreover, EH(S) is the smallest set that determines S up to maximality as the following result shows.

152

Rosales et al.

Proposition 12. Let S be a numerical semigroup and let {g1 , . . . , gt } ⊆ H(S). The following conditions are equivalent. (1) S is maximal (with respect to set inclusion) in the set of all numerical semigroups S  such that S  ∩ {g1 , . . . , gt } is empty. (2) EH(S) ⊆ {g1 , . . . , gt } . Proof. Let x ∈ EH(S) . By Proposition 9, S ∪ {x} is a numerical semigroup containing S properly. Thus if (1) holds, then (S ∪ {x}) ∩ {g1 , . . . , gt } = ∅ , which yields x ∈ {g1 , . . . , gt } . (2) implies (1) follows easily from Proposition 9 and Lemma 10. A numerical semigroup is irreducible if it cannot be expressed as an intersection of two numerical semigroups containing it properly. In [11] it is shown that S is irreducible if and only if S is maximal in the set of numerical semigroups not containing g(S) . Therefore we get the following consequence. Corollary 13. #EH(S) = 1.

A numerical semigroup S is irreducible if and only if

The set EH(S) is a subset of T(S) = {x ∈ H(S): x + s ∈ S for all s ∈ S\{0}} . The cardinality of this set is usually denoted by t(S) , and it is known as the type of S . Denote by Ap(S, m(S)) = {s ∈ S: s − m(S) ∈ S}, the Ap´ery set of S with respect to m(S) (see [1]). It is well known (see for instance [5]) that Ap(S, m(S)) = {0 = w(0), w(1), . . . , w(m(S) − 1)} , where w(i) is the least element in S congruent with i modulo m(S) . Moreover if {w(i1 ), . . . , w(it )} = Maximals≤S (Ap(S, m(S))) , where a ≤S b if b − a ∈ S , then T(S) = {w(i1 ) − m(S), . . . , w(it ) − m(S)} . This in particular implies that #EH(S) ≤ t(S) ≤ m(S) − 1. A numerical semigroup S is of maximal embedding dimension if e(S) = m(S). We prove next that if the bound given in the above inequality is reached for a numerical semigroup S , then S must be of maximal embedding dimension. Proposition 14. Let S be a numerical semigroup with m = m(S) and #EH(S) = m − 1 . Then S is a maximal embedding dimension numerical semigroup. Proof. We already know that Ap(S, m) = {0, w(1), . . . , w(m − 1)} . We show that {m, w(1), . . . , w(m − 1)} is a minimal system of generators of S . Since s + s ∈ Ap(S, m) implies that both s and s are in Ap(S, m) , in order to prove this, it suffices to show that for every i, j ∈ {1, . . . , m − 1} ,

Rosales et al.

153

w(i) + w(j) ∈ Ap(S, m) . Besides, as #EH(S) = m − 1 , we have that EH(S) = {w(1) − m, . . . , w(m − 1) − m} , whence w(i) + (w(j) − m) ∈ S for all i, j , meaning that w(i) + w(j) ∈ Ap(S, m). Example 15. The semigroup S = 5, 11, 12, 13, 14 is a maximal embedding dimension numerical semigroup with EH(S) = {6, 7, 8, 9} . Hence #EH(S) = 4 = m(S) − 1 . The converse of Proposition 14 is not true. The maximal embedding dimension numerical semigroup S = 5, 6, 7, 8, 9 has EH(S) = {3, 4} and thus #EH(S) = 2 < 5 − 1 . Lemma 16. Let S be a numerical semigroup and let {g1 , . . . , gt } ⊆ H(S). The following conditions are equivalent. (1) S is maximal in the set of numerical semigroups S  such that S  ∩ {g1 , . . . , gt } is empty. (2) If x ∈ H(S), then there exist i ∈ {1, . . . , t} and k ∈ N\{0} such that gi − kx ∈ S . Proof. (1) implies (2). Let x ∈ H(S) . Since S  S, x, we have that S, x ∩ {g1 , . . . , gt } = ∅ . Hence there exist i ∈ {1, . . . , t} , k ∈ N\{0} and s ∈ S such that gi = s + kx , whence gi − kx ∈ S . (2) implies (1). Let S  be a numerical semigroup such that S  S  . Take  x ∈ S \S . Then S  S, x ⊆ S  and by hypothesis there exist i and k such that gi − kx ∈ S , whence gi ∈ S, x, which leads to gi ∈ S  . In [3] the following construction is proposed to find a symmetric numerical semigroup containing a given one. Recall that a numerical semigroup S is symmetric if for every x ∈ Z\S , g − x ∈ S . Let S be a numerical semigroup with g(S) odd. If S is not symmetric, then there exists x ∈ H(S) such that g − x ∈ H(S) . Let h be maximum fulfilling this condition. Then S ∪ {h} turns out to be a numerical semigroup containing S . We perform the same procedure on S ∪ {h} and in a finite number of steps we will find S  symmetric with S ⊆ S  . The following result can be viewed as a generalization of this characterization and construction. Proposition 17. If there exists

Let S be a numerical semigroup and {g1 , . . . , gt } ⊆ H(S).

h = max{x ∈ H(S): 2x ∈ S, gi − x ∈ S for all i ∈ {1, . . . , t}}, then S ∪ {h} is a numerical semigroup (and (S ∪ {h}) ∩ {g1 , . . . , gt } = ∅ ). Proof. Clearly 2h ∈ S . If for some s ∈ S\{0} we have that h + s ∈ S , since 2(h + s) ∈ S and h < h + s, we get that gi − (h + s) ∈ S for some i. But this yields gi − h ∈ S , in contradiction with the definition of h .

154

Rosales et al.

Corollary 18. Let S be a numerical semigroup and {g1 , . . . , gt } ⊆ H(S). The following conditions are equivalent. (1) S is maximal in the set of all numerical semigroups S  with S  ∩ {g1 , . . . , gt } = ∅ . (2) For every x ∈ N , if x ∈ H(S) and 2x ∈ S , then gi − x ∈ S for some i ∈ {1, . . . , t} . Proof. (1) implies (2) follows from Proposition 17. For the converse, in view of Lemma 16, it suffices to show that for every x ∈ H(S) , there exists appropriate i and k such that gi −kx ∈ S . Let x ∈ H(S) and set k = max{n ∈ N\{0}: nx ∈ S} . Clearly kx ∈ H(S) and 2kx ∈ S . By hypothesis gi − kx ∈ S for some i ∈ {1, . . . , t} . As a consequence of these two results, we obtain a characterization of irreducible numerical semigroups that gathers the characterizations known for symmetric and pseudo-symmetric numerical semigroups (see [2] for the definition of pseudo-symmetric numerical semigroup). Corollary 19. Let S be a numerical semigroup. Then S is irreducible if and only if for all x ∈ N , x ∈ H(S) and 2x ∈ S implies g(S) − x ∈ S . Proof. As we pointed in the Introduction, S is an irreducible numerical semigroup if and only if S is maximal with the constrain g = g(S) ∈ S (in [2], [3] it is shown that a numerical semigroup S is symmetric (respectively pseudosymmetric) if and only if g(S) is odd (respectively even) and it is maximal in the set of numerical semigroups not containing its Frobenius number; in [11] it is shown that a numerical semigroup is irreducible if and only if it is maximal in the set of numerical semigroups not containing its Frobenius numbers). Using Corollary 18 the proof follows easily. Compare this characterization with the characterizations given in [2], [3] for symmetric and pseudo-symmetric numerical semigroups. Corollary 19 covers both classes of numerical semigroups. If we want to check whether or not a numerical semigroup S is symmetric (respectively pseudo-symmetric), we can use Corollary 19 imposing that g(S) is odd (respectively even). Let {g1 , . . . , gt } ⊂ N . Define S(g1 , . . . , gt ) = {S  ∈ S: S  ∩ {g1 , . . . , gt } = ∅}. Proposition 17 can be used to find a maximal element in S(g1 , . . . , gt ) . We only have to take as starting point S = {0, max{g1 , . . . , gt } + 1, →} and define recursively • S0 = S ,

Rosales et al.

155

• Sn+1 = Sn ∪ {h(Sn )} , where h(Sn ) = max{x ∈ H(Sn ): 2x ∈ Sn , gi − x ∈ Sn for all i ∈ {1, . . . , t}}; if h(Sn ) does not exist, then Sn is the desired semigroup (Corollary 18, gives this stop condition). Example 20.

We compute an element in Maximals⊆ (S(5, 6)) .

1. S0 = S = {0, 7, →}, h(S0 ) = 4 , 2. S1 = {0, 4, 7, →} , Maximals⊆ (S(5, 6)).

h(S1 ) is not well defined and thus S1

∈

Remark 21. In [7] it is shown that S(g1 , . . . , gt ) has a maximum (with respect to set inclusion) if and only if N\D(g1 , . . . gt ) is a numerical semigroup (actually this is the maximum), where D(g1 , . . . , gt ) is the set of positive integers dividing gi for some i ∈ {1, . . . , t} . In the above example, N\D(5, 6) = S1 = max⊆ (S(5, 6)) . In general there are several maximal numerical semigroups in S(g1 , . . . , gt ) . Note also that the above construction ensures that for any finite subset X of N , we obtain that S(X) = ∅ and therefore Maximals⊆ (S(X)) = ∅ . 5. The decomposition into irreducibles of a numerical semigroup In this section we use the tools described so far to compute a minimal decomposition of a numerical semigroup in terms of irreducibles. If S is a numerical semigroup, we already know how to calculate the finite variety V(S) with the procedure explained after Lemma 10. Denote by I(S) = {S  ∈ V(S): S  is irreducible}. Every numerical semigroup can be expressed  as an¯ intersection of irreducible numerical semigroups. Then clearly S = S∈I(S) S . And in this intersection ¯ we can remove those numerical semigroups that are not minimal with respect to set inclusion. Thus we obtain the following result. Proposition 22. Let S be a numerical semigroup and let I(S) be defined as above. Assume that Minimals⊆ (I(S)) = {S1 , . . . , Sn } . Then S = S1 ∩ · · · ∩ Sn . This decomposition does not have to be minimal (in the sense of minimal number of irreducibles involved) as the following example shows. Example 23. Let S be the semigroup generated by 5, 6, 8. Using Proposition 9 and Corollary 13, we compute now the set Minimals(I(S)) . Since

156

Rosales et al.

EH(S) = {7, 9} , by Proposition 9, S ∪ {7} and S ∪ {9} are numerical semigroups. In this case we have that since EH(S ∪{7}) = {9} , S ∪{7} is irreducible (Corollary 13), which implies that it belongs to Minimals⊆ (I(S)) . The semigroup S ∪ {9} is not irreducible (the set EH(S ∪ {9}) is equal to {3, 4, 7} ). By Proposition 9 the sets S ∪ {9, 3} , S ∪ {9, 7} and S ∪ {9, 4} are also numerical semigroups. It is easy to check that S∪{9, 3} and S∪{9, 4} are irreducible semigroups and that S ∪ {9, 7} contains the semigroup S ∪ {7} (the first irreducible we found). Hence the set Minimals⊆ (I(S)) = {S ∪ {7}, S ∪ {9, 3}, S ∪ {9, 4}}. Finally, S = (S ∪ {7}) ∩ (S ∪ {9, 4}) ∩ (S ∪ {9, 3}) = (S ∪ {7}) ∩ (S ∪ {9, 4}). However, if we are looking for the least n such that S = S1 ∩ · · · ∩ Sn , with S1 , . . . , Sn ∈ I(S) , then it suffices to search among the decompositions with elements in Minimals⊆ (I(S)) . Proposition 24. Let S be a numerical semigroup. If S = S1 ∩ · · · ∩ Sn with S1 , . . . , Sn ∈ I(S), then there exists S1 , . . . , Sn ∈ Minimals⊆ (I(S)) such that S = S1 ∩ · · · ∩ Sn . Proof. For every i ∈ {1, . . . , n} , if Si does not belong to Minimals⊆ (I(S)) , then take Si ∈ Minimals⊆ (I(S)) such that Si ⊆ Si . The next proposition sheds some light on which semigroups are required in a decomposition. Proposition 25. Let S be a numerical semigroup and let S1 , . . . , Sn ∈ V(S). The following conditions are equivalent. (1) S = S1 ∩ · · · ∩ Sn . (2) For all h ∈ EH(S), there exists i ∈ {1, . . . , n} such that h ∈ Si . Proof. (1) implies (2). If h ∈ EH(S) , then h ∈ S and thus h ∈ Si for some i ∈ {1, . . . , n} . (2) implies (1). If S  S1 ∩· · ·∩Sn , then by Proposition 9 and Lemma 10 h = max((S1 ∩ · · · ∩ Sn )\S) is in EH(S) , and in all the Si , in a contradiction with the hypothesis. We can compute Minimals⊆ (I(S)) = {S1 , . . . , Sn } . {1, . . . , n} , set C(Si ) = {h ∈ EH(S): h ∈ Si }.

For every i ∈

By Proposition 25 we know that S = Si1 ∩ · · · ∩ Sir if and only if C(Si1 ) ∪ · · · ∪ C(Sir ) = EH(S).

Rosales et al.

157

From the above results we can obtain a method for computing a decomposition of S as an intersection of irreducible semigroups using the least possible number of them. Algorithm 26.

Let S be a non-irreducible semigroup.

(1) Compute the set EH(S). (2) Set I = ∅ and C = {S} . (3) For all S  ∈ C , compute (using Proposition 9) all the semigroups S¯ such ¯  ) = 1. Remove S  from C . Let B be the set formed by the that #(S\S semigroups constructed in this way. (4) Remove from B the semigroups S  fulfilling that EH(S) ⊆ S  . (5) Remove from B the semigroups S  such that there exists S˜ ∈ I with S˜ ⊆ S  . (6) Set C = {S  ∈ B | S  is not irreducible} . (7) Set I = I ∪ {S  ∈ B | S  is irreducible} . (8) If C = ∅ go to Step 3. (9) For every S ∈ I , compute C(S). (10) Choose {S1 , . . . , Sr } such that r is minimum fulfilling that C(S1 ) ∪ · · · ∪ C(Sr ) = EH(S). (11) Return S1 , . . . , Sr . Next we illustrate this method with an example. Example 27. We take again the semigroup S = 5, 6, 8. We have that EH(S) = {7, 9}. Performing the steps of the above algorithm we get (in the Steps 6 and 7) that I = {5, 6, 7, 8} and C = {5, 6, 8, 9}}. Since C = ∅ , we go back to Step 3 obtaining that I = {5, 6, 7, 8, 3, 5, 4, 5, 6} and C = ∅ . Step 8 yields C(5, 6, 7, 8) = {9}, C(3, 5) = {7}, C(4, 5, 6) = {7}. The minimal decompositions of S are S = 5, 6, 7, 8 ∩ 3, 5 and S = 5, 6, 7, 8 ∩ 4, 5, 6. Acknowledgments The authors would like to thank the referee for her/ his suggestions and comments. This paper was supported by the project BFM2000-1469.

158

Rosales et al. References

[1] Ap´ery, R., Sur les branches superlin´eaires des courbes alg´ebriques, C. R. Acad. Sci. Paris 222 (1946), 1198–1200. [2] Barucci, V., D. E. Dobbs, and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Memoirs of the Amer. Math. Soc. 598 (1997). [3] Fr¨ oberg, R., G. Gottlieb, and R. H¨ aggkvist, On numerical semigroups, Semigroup Forum 35 (1987), 63–83. [4] Kunz, E., The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1973), 748–751. [5] Rosales, J. C., On numerical semigroups, Semigroup Forum 52 (1996), 307–318. [6] Rosales, J. C. and P. A. Garc´ıa-S´ anchez, “Finitely Generated Commutative Monoids”, Nova Science Publishers, New York, 1999. [7] Rosales, J. C., P. A. Garc´ıa-S´ anchez, J. I. Garc´ıa-Garc´ıa, and J. A. Jim´enez Madrid, Fundamental gaps in numerical semigroups, submitted. [8] Rosales, J. C. and M. B. Branco, Irreducible numerical semigroups with arbitrary multiplicity and embedding dimension, J. Algebra (to appear). [9] Rosales, J. C. and M. B. Branco, Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups, J. Pure Appl. Algebra 171 (2002), 303–314. [10] Rosales, J. C. and M. B. Branco, Decomposition of a numerical semigroup as an intersection of irreducible numerical semigroups, B. Belg. Math. SocSim. 9 (2002), 373–381. [11] Rosales, J. C. and M. B. Branco, Irreducible numerical semigroups, Pacific J. Math. (to appear).

´ Departamento de Algebra Universidad de Granada E-18071 Granada, Spain [email protected], [email protected], [email protected]

Received September 20, 2002 and in final form January 20, 2003 Online publication February 28, 2003

Instituto Andaluz de Astrof´ısica c/ Camino Bajo de Hu´etor, 24 E-18080 Granada, Spain [email protected]