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On Numerical Semigroups with High Embedding Dimension
203, 567]578 Ž1998. JA977341
JOURNAL OF ALGEBRA ARTICLE NO.
On Numerical Semigroups with High Embedding Dimension J. C. Rosales and P. A. Garcıa-Sanchez ´ ´ ´ Departamento de Algebra, Facultad de Ciencias, Uni¨ ersidad de Granada, 18071, Granada, Spain E-mail: [email protected] and [email protected] Communicated by Walter Feit Received August 7, 1997
We compute the number of elements of a minimal system of generators for the congruence of a numerical semigroup with embedding dimension minus multiplicity equal to zero, one, and two. For symmetric numerical semigroups we study the cases of embedding dimension minus multiplicity equal to one, two, and three. Q 1998 Academic Press
INTRODUCTION A numerical semigroup is a subset S of N closed under addition, containing the zero element and generating Z as a group. From this definition, one obtains that S has a minimal Žwith respect to the inclusion. system of generators as a semigroup, n 0 - n1 - ??? - n p 4 , verifying that the greatest common divisor of these generators is one. The number p q 1 is usually called the embedding dimension of S and the number n 0 is called the multiplicity of S. We define the semigroup homomorphism w : N pq 1 ª N as
w Ž a0 , a1 , . . . , a p . s a0 n 0 q a1 n1 q ??? qa p n p . Let us denote by s the kernel congruence of w . Thus, S ( N pq 1rs . Redei ´ shows in w3x that s is finitely generated. The scope of this paper is the computation of the number of elements of a system of generators for s with minimal cardinality. This problem has been already studied for some cases: Herzog studies in w2x this problem for the case p s 1 and p s 2; 567 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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Bresinsky treats in w1x the case when p s 3 and S is symmetric; Rosales studies in w5x the case p s n 0 y 1 and in w6x the case p s n 0 y 2 and S symmetric. The main purpose of this paper is the study of the cases p s n 0 y 2 and p s n 0 y 3 for S an arbitrary numerical semigroup and the cases p s n 0 y 3, and p s n 0 y 4 for S a symmetric numerical semigroup. In the first section, we obtain that if p s n 0 y 2, the number of elements in a system of generators for s with minimal cardinality is Ž n 0 y 1.Ž n 0 y 2.r2 y 1 or Ž n 0 y 1.Ž n 0 y 2.r2. We also determine the number of elements in a minimal presentation for S when p s n 0 y 3. In this case, we see that this number is between Ž n 0 y 2.Ž n 0 y 3.r2 y 2 and Ž n 0 y 2.Ž n 0 y 3.r2. As an application of the mentioned results, we give a classification of the semigroups with n 0 F 6 by the number of elements in a system of generators of its associated congruence with minimal cardinality. In the second section, we study the symmetric case. For p s n 0 y 2, the first author showed in w6x that ar s Ž n 0 y 1.Ž n 0 y 2.r2 y 1. We extend this result for the cases p s n 0 y 3 and p s n 0 y 4, and we show that the cardinality of a minimal system of generators for the congruence associated to the semigroup is Ž n 0 y 2.Ž n 0 y 3.r2 y 1 and Ž n 0 y 3.Ž n 0 y 4.r 2 y 1, respectively. As an application we study the cases in which S can be a complete intersection and we compute the number of a minimal system of generators for the congruence associated to a symmetric numerical semigroup with n 0 F 8. PRELIMINARIES In w4x, the first author gives an algorithm to compute a system of generators, r , for s with minimal cardinality. From the results given in that paper, it is derived that the concepts of a system of generators for s with minimal cardinality and a minimal system Žwith respect to the inclusion. of generators of s coincide. Next, we give a sketch of this construction. For every n g S, we define the graph G n s Ž Vn , En ., as Vn s n i g n 0 , . . . , n p 4 : n y n i g S 4 , En s w n i , n j x : n y Ž n i q n j . g S, i / j g 0, . . . , p 4 4 . We define rn as Ž1. If G n is not connected and G1n s Ž Vn1 , En1 ., . . . , G nr s Ž Vnr , Enr . are its connected components, then for every 1 F i F r we select an element
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a i s Ž a0 i , . . . , a p i . g N pq 1 _ 04 such that w Ž a i . s n and a k i s 0 for all n k i f Vni. Define rn s Ž a 1 , a 2 ., Ž a 1 , a 3 ., . . . , Ž a 1 , a r .4 . Ž2. If G n is connected, we define rn s B. The set r s Dng N rn is a system of generators for s with minimal cardinality Žsee w4x.. Given 0 / n g S, we define SŽ n. to be the Apery ´ set of n S Ž n . s s g S: s y n f S 4 . These sets are finite. In w4x, it is shown that if G n is not connected, then there exist s g SŽ n 0 . _ 04 and i g 1, . . . , p4 such that n s s q n i . The conductor of a numerical semigroup is the maximum integer not belonging to the semigroup. Such an element always exists, due to the fact that the group generated by a numerical semigroup is Z. It is also known that if x is the greatest element in S Ž n 0 . then the conductor, C, of S is C s x y n 0 . In w5x, the first author studies the relationship of a minimal set of generators for the congruence associated to a semigroup and a minimal set of generators for the congruence associated to the semigroup obtained adding its conductor to the given semigroup, provided that the conductor is greater than n 0 . ŽThe case C - n 0 only occurs when the semigroup is of the form ² n 0 , n 0 q 1, . . . , 2 n 0 y 1:. We may assume that the semigroup under study is not of this type, because this kind of semigroup can be studied using the results appearing in w5x.. Let us denote by S1 s S j C 4 . Then, S1 is generated by A1 s n 0 , . . . , n p , C 4 . The set A1 is a minimal system of generators for S1 if and only if x g SŽ n 0 . _ 0, n1 , . . . , n p 4 . If the set A1 is not a minimal system of generators for S1 , then it can be shown that ar s ar 1 , where r 1 is a minimal system of generators for the congruence associated to S1. Thus, if we are looking for bounds for ar , it does not matter if we study S1 instead of S. Let AX1 be a minimal system of generators for S1 and let S2 s S1 j C14 , with C1 the conductor of S1. Then, S ; S1 ; S2 and we can check once more if AX1 j C14 is a minimal generating system for S2 . If it is not so, we construct S3 . We continue this process until the semigroup constructed is minimally generated by the minimal system of generators plus the conductor of the previous semigroup. This procedure must stop, since it can be shown that there is a positive integer m such that Sm s S n 0 s ² n 0 , n 0 q 1, . . . , 2 n 0 y 1: s n 0 q N and in this case AXmy 1 j Cmy14 is a minimal system of generators for S m . Thus, we can assume that S s ² n 0 , . . . , n p : and S1 is minimally generated by n 0 , . . . , n p , C 4 . From the results presented by the first author in w5x, it is derived that r 1 has as many elements as r plus p q 2 elements Žcorresponding to the graphs G Cq n , . . . , G Cqn , G 2 C . minus the elements 0 p
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which arise when there is no path from n 0 to n i in the graphs of the form G xqn i , where x is the greatest element in SŽ n 0 . and i g 1, . . . , p4 . This idea will enable us to use the results known for semigroups with embedding dimension equal to the multiplicity of the semigroup in order to achieve bounds for the number of elements in minimal presentations of semigroups with embedding dimension their multiplicity minus one. Once this bound is obtained, we can use the same argument to compute a bound for semigroups with embedding dimension their multiplicity minus two. In each step, from the given semigroup we construct a new semigroup being under conditions already studied and then, we count how many elements n i verify that n i and n 0 belong to different connected components of G xqn i , where x is the greatest element in SŽ n 0 .. Finally, we recall the definition of two classes of semigroups. The semigroup S is symmetric if the set SŽ n. has a maximum with respect to the ordering induced by the addition in S. In this case, the semigroup ring K w S x s [s g S Kys is Gorenstein. The semigroup S is a complete intersection if ar is exactly p, which is the minimum possible number of elements that a system of generators for s may contain Žsee w2x..
1. NUMERICAL SEMIGROUPS WITH HIGH EMBEDDING DIMENSION For a numerical semigroup, the number of elements in SŽ n 0 . is exactly n 0 . Note that all the generators n i , i ) 0, are in S Ž n 0 .. Thus, the maximal embedding dimension for a semigroup is exactly n 0 Žfor the zero element is always in SŽ n 0 ... The semigroups having maximal embedding dimension have been studied by the first author in w5x. In the mentioned paper, these kinds of semigroups are called MED-semigroups, and it is shown that ar s n 0 Ž n 0 y 1.r2. An example of this kind of semigroups is the semigroups of the form S s ² n 0 , n 0 q 1, . . . , 2 n 0 y 1:, which already appeared in the preliminaries. 1.1. Numerical Semigroups with Embedding Dimension Their Multiplicity Minus One In this section, we study the minimal sets of generators for the congruences associated to numerical semigroups, S, such that S Ž n 0 . s 0, n1 , . . . , n p , w 4 , that is to say, numerical semigroups with embedding dimension p q 1 s n 0 y 1.
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Note that, since 0 / w g SŽ n 0 ., w can be written as w s a1 n1 q ??? qa p n p , with a i / 0 for some i. Thus, w y n i g SŽ n 0 .. Therefore w s n i q n j for some i, j g 1, . . . , p4 Žnot necessarily different.. Let S1 s ² n 0 , . . . , n p , C :. As we have seen in the preliminaries, we can assume that S1 is minimally generated by n 0 , . . . , n p , C 4 . Note that this is equivalent to the fact that w is the greatest element in SŽ n 0 ., because if n i is the greatest element in SŽ n 0 . for some i, then n i s Ž n i y n 0 . q n 0 s C q n 0 and therefore n 0 , . . . , n p , C 4 does not minimally generate S1. Hence, S1 is a MED-semigroup and ar 1 s n 0 Ž n 0 y 1.r2. In order to count the elements belonging to r , we must count how many generators n i , 1 F i F p verify that n i and n 0 are in different connected components of G wqn i . Let w s n i q n j . If n k and n 0 are not in the same connected component of G wq n k for some k g 1, . . . , p4 , then w q n k y Ž n i q n 0 . f S and w q n k y Ž n j q n 0 . f S. This implies that n j q n k , n i q n k g SŽ n 0 . which leads to n i s n j s n k . Thus, the only graph that can add a new element to r is G wq n i when w s 2 n i and w cannot be expressed in another way. Let us show that, as a matter of fact, in this case n i and n 0 are in different connected components of G wq n i . If it were not so, there would exist a path from n i to n 0 , which would mean that w q n i y Ž n i q n k . g S for some k / i. Nevertheless, w q n i y Ž n i q n k . s 2 n i y n k s w y n k and if w y n k g S, then w must admit an expression different from 2 n i , a contradiction. We get the following theorem: THEOREM 1. Let S s ² n 0 , . . . , n p : be a numerical semigroup such that SŽ n 0 . s 0, n1 , . . . , n p , w4 and let r be a minimal system of generators for the congruence associated to S. Ž1.
If w s n i q n j for some i, j g 1, . . . , p4 , i / j then ar s
Ž2.
Ž n0 y 1. Ž n0 y 2. 2
y 1.
Otherwise, ar s
Ž n0 y 1. Ž n0 y 2. 2
.
Proof. Let q be the number of elements in r arising from the fact that n i is not in the same connected component of n 0 in G wqn i . We have just shown that if w s n i q n j with i / j, then q s 0. Otherwise, q s 1. This concludes the proof, because ar 1 s n 0 Ž n 0 y 1.r2 s ar q Ž p q 2. y q s ar q n 0 y q and therefore ar s Ž n 0 y 1.Ž n 0 y 2.r2 y 1 q q.
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As an application of the previous theorem, one can get a classification of semigroups having n 0 F 5 by the number of elements in a minimal system of generators for its congruence. The only cases which are not complete intersections and are not covered by w2x or w5x are n 0 s 4, p s 2 and n 0 s 5, p s 3. These are covered by Theorem 1. 1.2. Numerical Semigroups with Embedding Dimension Their Multiplicity Minus Two In this case, S Ž n 0 . s 0, n1 , . . . , n p , w 1 , w 2 4 . As in the previous section, we can assume that S s ² n 0 , . . . , n p : and that S1 is minimally generated by n 0 , . . . , n p , C 4 , that is to say, the greatest element in S Ž n 0 . is in w 1 , w 2 4 . We will suppose that w 2 is such a maximum. Hence, S1 s ² n 0 , . . . , n p , C : verifies that S1Ž n 0 . s 0, n1 , . . . , n p , C, w 14 and therefore ar 1 g
Ž n0 y 1. Ž n0 y 2. 2
y 1,
Ž n0 y 1. Ž n0 y 2. 2
.
Thus, we are interested in the elements n i such that n 0 and n i are in different connected components of G w 2qn i . Let q be the number of such elements. We have ar s ar 1 y n 0 q 1 q q. The following cases are possible: Case 1. aVw 2 G 3. Take n i , n j , n k as three elements in Vw 2 . If n t and n 0 are in a different connected component of G w 2qn t then n i q n t , n j q n t , n k q n t g SŽ n 0 . _ 0, n1 , . . . , n p 4 Žif, for instance, n i q n t y n 0 g S, take n s g Vw 2yn i ; then w 2 q n t y Ž n s q n 0 . s Ž w 2 y n i y n s . q Ž n i q n t y n 0 . g S .. But this is impossible, because aSŽ n 0 . _ 0, n1 , . . . , n p 4 s 2. Thus, in this case q s 0 and Ža. If aVw G 2 then 1 ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 2.
Žb. If aVw s 1 then 1 ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
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Case 2. aVw 2 s 2. Let Vw 2 s n i , n j 4 . From the fact that aSŽ n 0 . _ 0, n1 , . . . , n p 4 s 2 it is easily derived that w 2 must be equal to n i q n j . If n k is such that there is no path from n 0 to n k in G w 2qn k, then n i q n k , n j q n k g SŽ n 0 .. Hence, n i s n k and w 1 s 2 n i , or n j s n k and w 1 s 2 n j . Let us study the different possibilities for aVw 1. Ža. aVw s 1. ŽNote that in this case w 1 s 2 n t for some t and this 1 expression is unique.. Ži. If w 1 / 2 n i and w 1 / 2 n j then n k and n 0 are in the same connected component of G w 2qn k, for all k. Hence, q s 0 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
Žii. If w 1 s 2 n i then this expression is unique. Hence, w 2 q n i y Ž n i q n k . g S if and only if k g i, j4 and w 2 q n i y Ž n j q n k . g S if and only if k s i, which means that the connected component of n i in G w 2qn i is n i , n j 4 . Thus, q s 1 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
.
Žiii. If w 1 s 2 n j we obtain the same result. Žb. aVw G 2. 1 Ži. If w 1 / 2 n i and w 1 / 2 n j then n k and n 0 are in the same connected component of G w 2qn k, for all k. Hence, q s 0 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 2.
Žii. If w 1 s 2 n i then, since aVw G 2, w 1 can be expressed as 1 n k q n l with i f k, l 4 . Let us show that n i and n 0 are in the same connected component of G w 2qn i . Note that w 2 q n i y Ž n j q n k . s 2 n i y n k s n l g S and that w 2 q n i y Ž n j q n l . s 2 n i y n l s n k g S. Thus, n i , n j , n k , n l 4 are in the same connected component of G w qn . Since 2 i i f k, l 4 then n j q n k and n j q n l are different from w 2 . This means that n j q n l f SŽ n 0 . or n j q n k f SŽ n 0 . Žor both.. In the first case, w 2 q n i y Ž n k q n 0 . s n j q n l y n 0 g S and in the second, w 2 q n i y Ž n l q n 0 . s n j q n k y n 0 g S, which implies in both cases that n 0 is connected with n i . Thus, in this case q s 0 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 s
Ž n0 y 2. Ž n0 y 3.
Žiii. If w 1 s 2 n j we get the same result.
2
y 2.
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Case 3. aVw 2 s 1. Then, w 2 s 2 n i or w 2 s 3n i for an element i g 1, . . . , p4 . In both cases, the expression of w 2 is unique and therefore n i is an isolated point of G w 2qn i Žotherwise, w 2 q n i y Ž n i q n k . s w 2 y n k g S for some k / i which is impossible, because aVw 2 s 1.. Thus, q G 1. Ža. aVw s 1. If n k and n 0 are in different connected components of 1 G w 2qn k, then n i q n k g SŽ n 0 ., and this only occurs if i s k. Hence, q s 1 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
.
Žb. aVw s 2. If w 2 s 3n i then w 1 must be 2 n i and this expression 1 must be unique, since aVw 2 s 1. But this implies that aVw 1 s 1, which is a contradiction. Then, w 2 s 2 n i , w 1 s n k q n l , for some k / l, and these expressions are unique. If n t Ž t / i . and n 0 are not connected in G w 2qn t then w 2 q n t y Ž n i q n 0 . f S, which means that n i q n t g SŽ n 0 . and therefore i, t 4 s l, k 4 . Hence, there is no s f i, t 4 such that n s and n 0 are not connected in G w 2qn s. Let us show that if w 1 s n i q n t then n i , n t 4 is a connected component of G w 2qn t and therefore q s 2. The element w 2 q n t y Ž n i q n s . s w 1 y n s belongs to S if and only if s g i, t 4 , because w 1 is expressed in an unique way. Besides, w 2 q n t y Ž n t q n s . s w 2 y n s is in S only if s s i, because Vw 2 s n i 4 . Thus, ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 q 2 s
Ž n0 y 2. Ž n0 y 3. 2
.
Žc. aVw G 3. As before, w 2 must be equal to 2 n i . If n j Ž j / i . and 1 n 0 are in different connected components of G w 2qn j , then w 2 q n j y Ž n i q n 0 . f S and therefore n i q n j s w 1. Since aVw 1 G 3, we can choose n k g Vw 1 _ n i , n j 4 . Note that w 2 q n j s 2 n i q n j s w 1 q n i and consequently n i , n j , and n k are in the same connected component of G w 2qn j . Let us show that n k and n 0 are connected. The element w 2 q n j y Ž n k q n 0 . s w 1 q n i y Ž n k q n 0 . s Ž w 1 y n k . q n i y n 0 belongs to S if and only if w 1 y n k q n i f SŽ n 0 . Žnote that w 1 y n k g S .. It is enough to show that w 1 y n k q n i / w 1 and w 1 y n k q n i / w 2 . Since n i / n k , w 1 y n k q n i cannot be w 1. If w 1 y n k q n i s w 2 , then w 1 q n i s 2 n i q n k and therefore n i q n j s w 1 s n i q n k , which means that n j s n k , and this is not possible. Thus, q s 1 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
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We have shown the following theorem: THEOREM 2. Let S s ² n 0 , . . . , n p : be a numerical semigroup such that SŽ n 0 . s 0, n1 , . . . , n p , w 1 , w 2 4 and let r be a minimal system of generators for the congruence associated to S. Then ar g s
p Ž p q 1. 2
y 2,
p Ž p q 1.
Ž n0 y 2. Ž n0 y 3. 2
2 y 2,
Ž n0 y 2. Ž n0 y 3. 2
.
Furthermore, if we assume w 1 - w 2 , then ar s Ž n 0 y 2.Ž n 0 y 3.r2 y 2 if and only if aVw 2 G 2 and aVw 1 G 2. ar s Ž n 0 y 2.Ž n 0 y 3.r2 y 1 if and only if aVw 2 G 3 and aVw 1 s 1, or aVw 2 s 2, aVw 1 s 1, and Vw 2 l Vw 1 s B, or aVw 2 s 1 and aVw 1 G 3. ar s Ž n 0 y 2.Ž n 0 y 3.r2 if and only if aVw 2 s 2 and Vw 1 ; Vw 2 , or aVw 2 s 1 and aVw 1 F 2. v v
v
As we did in the previous section for n 0 g 1, . . . , 54 , one can now study the case n 0 s 6 and see which are the possible values of ar . The cases that are not complete intersections and do not lay in the scope of the results in w2x nor in the results appearing in w5x are p s 3 and p s 4. The former case is covered by Theorem 2 and the latter by Theorem 1. 1.3. Numerical Semigroups with Embedding Dimension Their Multiplicity Minus Three From the results obtained in the previous sections one could think that the bounds for this case are Ž n 0 y 3.Ž n 0 y 4.r2 y 3 F ar F Ž n 0 y 3.Ž n 0 y 4.r2. If we take S s ²7, 8, 10, 19:, we get SŽ7. s 0, 8, 10, 16, 18, 19, 204 and ar s 7 ) Ž7 y 3.Ž7 y 4.r2, and therefore these bounds are not correct. If our computations are correct, the number of elements in r is in the interval wŽ n 0 y 3.Ž n 0 y 4.r2 y 3, Ž n 0 y 3.Ž n 0 y 4.r2 q 1x. We do not include the proof here because of the big amount of cases and subcases arising. This ‘‘bad’’ result makes one think of alternative ways of study for numerical semigroups with not so high embedding dimension. 2. SYMMETRIC NUMERICAL SEMIGROUPS WITH HIGH EMBEDDING DIMENSION If S is a symmetric semigroup then the set S Ž n 0 . has a maximum with respect to the ordering induced in S by the addition, and this maximum cannot belong to a minimal system of generators for S, but for n 0 s 2.
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This implies that the maximal embedding dimension is exactly n 0 y 1. In w6x, the first author denotes this kind of semigroups by MEDSY-semigroups and shows that ar s Ž n 0 y 1.Ž n 0 y 2.r2 y 1, which coincides with the first case of Theorem 1, but for n 0 equal to two or three. Note that symmetric numerical semigroups always lie in the first case of Theorem 1 because if SŽ n 0 . s 0, n1 , . . . , n p , w4 has a maximum, then w must be this maximum and it must be of the form w s n i q n j for some i / j Žprovided p G 2.. If S is a complete intersection, then S is Gorenstein and therefore must lie in the first case. Under this setting, n 0 must be equal to 4 or equal to 1 Ž S s N.. Thus, for n 0 G 5 there are no complete intersection semigroups with p s n 0 y 2. If p s n 0 y 3, and SŽ n 0 . s 0, n1 , . . . , n p , w 1 , w 2 4 then w 2 , the maximum of SŽ n 0 ., must be w 1 q n i for some n i , which leads to w 1 s 2 n i Žif aVw 1 G 2 then take n k / n l g Vw 1; the elements n k q n i , n l q n i g SŽ n 0 . _ w 2 4 , which means that w 1 s n k q n i s n l q n i , a contradiction.. Since w 2 y n j g S for all j, we have that the expression w 2 s 3n i is not unique, and therefore if p G 3 Žthe cases p s 0 and p s 1 are trivial and the case p s 2 has been already studied by Herzog in w2x., then aVw 2 G 3 which means that ar s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
Note that if S is a complete intersection and p s n 0 y 3, then from the equation ar s pŽ p q 1.r2 y 1 s p we get n 0 s 5. Thus, if n 0 G 6, there are no complete intersection numerical semigroups with their multiplicity equal to their embedding dimension minus two. Finally, let us study the case p s n 0 y 4: S Ž n 0 . s 0, n1 , . . . , n p , w 1 , w 2 , w 3 4 . The following theorem provides a result analogous to the ones obtained so far for p ) n 0 y 4. THEOREM 3. Let S s ² n 0 , . . . , n p : be a symmetric numerical semigroup such that SŽ n 0 . s 0, n1 , . . . , n p , w 1 , w 2 , w 2 4 , p G 3, and let r be a minimal system of generators for the congruence associated to S. Then ar s
Ž n0 y 3. Ž n0 y 4. 2
y 1.
Proof. Note that if S is symmetric then the greatest element x in SŽ n 0 . is in w 1 , w 2 , w 34 , because x y y g S for all y g SŽ n 0 .. Without loss of generality we can assume that x s w 3 and that w 1 - w 2 . Thus, S1 s S j
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C 4 is minimally generated by n 0 , . . . , n p , C 4 and S1 is under the conditions of the previous section. As we have done before, we must check how many elements in r arise from the fact that n i and n 0 are in different connected components of G w 3qn i for n i g 1, . . . , p4 . Assume that n i , i G 1, is such that n i and n 0 are in different connected components of G w 3qn i . Then, since w 3 y n k g S for all k g 1, . . . , p4 we get that if n 0 is not connected with n i in G w 3qn i , then the elements n k q n i are in SŽ n 0 . for all k g 1, . . . , p4 Žotherwise the edges w n i , n k x and w n k , n 0 x are in EŽG w 3qn i ... But p G 3 and n1 q n i , . . . , n p q n i are all different and non-comparable in S. This is a contradiction with the fact that the maximum number of non-comparable elements in S Ž n 0 . _ 0, n1 , . . . , n p 4 is two. This means that ar s ar 1 y n 0 q 2. If we want to show that ar s Ž n 0 y 3.Ž n 0 y 4.r2 y 1, then we must prove that ar 1 s Ž n 0 y 2.Ž n 0 y 3.r2 and this is equivalent to demonstrating Žby Theorem 2. that aVw 2 s 2 and Vw 1 ; Vw 2 , or that aVw 2 s 1 and aVw 1 F 2. First of all, let us show that aVw 2 F 2 and that aVw 1 F 2. If n i , n j , n k are three different elements in Vw 2 then take n l g Vw 3yw 2 . The elements n l q n i , n l q n j , n l q n k are three different non-comparable Žw.r.t. the ordering induced in S by the addition. elements in S Ž n 0 . _ 0, n1 , . . . , n p 4 , and this is not possible. In the same way, it is shown that aVw 1 F 2. Observe that w 3 y w 2 g SŽ n 0 ., and therefore w 3 y w 2 s w 2 , w 3 y w 2 s w 1 or w 3 y w 2 g n1 , . . . , n p 4 . If w 3 y w 2 s w 2 , then take n k g Vw 2 . The element w 2 q n k g SŽ n 0 . is greater than w 1 and it is neither w 2 nor w 3 , a contradiction. Analogously, we can show that w 3 y w 2 / w 1. Thus, w 3 y w 2 g n1 , . . . , n k 4 . We have seen that w 3 y w 1 / w 2 and therefore w 3 y w 1 s w 1 or w 3 y w 1 g n1 , . . . , n p 4 . If w 3 y w 1 s w 1 , take n k g Vw 1. Then, w 1 q n k g SŽ n 0 . _ w 3 , w 1 4 and therefore w 1 q n k s w 2 . Briefing, we get that there are two possible cases: Ž1. There exists i / j such that w 3 s w 2 q n i s w 1 q n j . Ž2. There exists i, j such that w 3 s w 2 q n i and w 2 s w 1 q n j . Let us study each case separately: Ž1. It is enough to show that if aVw s 2 then Vw ; Vw , because if 2 1 2 aVw 2 s 1 then, since aVw 1 F 2, we are done. Assume that aVw 2 s 2. Then w 2 s n s q n t for some s - t. As w 3 s w 2 q n i , then n s q n i , n t q n i g SŽ n 0 . _ w 34 . Hence, n s q n i s w 1 and n t q n i s w 2 s n s q n t , which means that n s s n i and consequently w 1 s 2 n i . Besides, w 3 s w 1 q n j s
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2 n i q n j s w 2 q n i and this leads to w 2 s n i q n j Ž i, j4 s s, t 4.. Let us assume that w 1 can be expressed in a different way. Then w 1 s n k q n l with i f k, l 4 and k / l. Since w 3 s w 1 q n j , n k q n j , n l q n j g SŽ n 0 .. However, the expression w 2 s n i q n j is unique, which implies that w 1 s n k q n j s n l q n j , and this is impossible since k / l. We have shown that Vw 1 s n i 4 ; n i , n j 4 s Vw 2 . Ž2. If we show that aVw s 1 then, since aVw F 2, we are done. If 2 1 aVw 2 s 2, then w 2 s n k q n l , for some n k - n l . Hence, n k q n i , n l q n i g SŽ n 0 . _ w 34 , because w 3 s w 2 q n i . This implies that n k q n i s w 1 and n l q n i s w 2 , but this leads to n l q n i s w 2 s w 1 q n j s n k q n i q n j , which means that n l s n k q n j , and this is impossible. The restriction p G 3 is unimportant, because for the cases p s 0 and p s 1 the semigroup is a complete intersection, and the case p s 2 has been already studied by Herzog for S a numerical semigroup not necessarily symmetric. With the results obtained in this paper one can compute the possible values of ar for the symmetric numerical semigroups with n 0 F 8. Observe that if n 0 F 6, the study can be done using the results exposed in the previous sections together with the results appearing in w2, 5x. Recall that if n 0 G 3 and the semigroup is symmetric, then p F n 0 y 2. Note also that, due to the results obtained by Herzog in w2x, if p s 2 then the semigroup is symmetric if and only if it is a complete intersection. Thus, for n 0 s 7 and p F 2 we get complete intersections; for p s 4 we can apply Theorem 3; for p s 5 we get a MEDSY-semigroup. Finally, if n 0 s 8 and p F 2 we get once more a complete intersection semigroup; for p g 4, 5, 64 we can use the results exposed in this paper; for p s 3, using the results obtained by Bresinsky in w1x, we get that ar g 3, 54 . Note that for n 0 s 8 and p s 3 s n 0 y 5 the statement ar s Ž n 0 y 4.Ž n 0 y 5.r2 y 1 is false. Taking, for example, S s ²8, 10, 12, 15:, one can compute r and obtain that ar s 3 / 4 = 3r2 y 1 s 5. REFERENCES 1. H. Bresinsky, Symmetric semigroups of integers generated by 4 elements, Manuscripta Math. 17 Ž1975., 206]219. 2. J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 Ž1970., 175]193. 3. L. Redei, ‘‘The Theory of Finitely Generated Commutative Semigroups,’’ Pergamon, ´ Elmsford, NY, 1965. 4. J. C. Rosales, An algorithmic method to compute a minimal relation for any numerical semigroup, Internat. J. Algebra Comput. 6, No. 4 Ž1996., 441]455. 5. J. C. Rosales, On numerical semigroups, Semigroup Forum 52 Ž1996., 307]318. 6. J. C. Rosales, On symmetric numerical semigroups, J. Algebra 182 Ž1996., 422]434.
JOURNAL OF ALGEBRA ARTICLE NO.
On Numerical Semigroups with High Embedding Dimension J. C. Rosales and P. A. Garcıa-Sanchez ´ ´ ´ Departamento de Algebra, Facultad de Ciencias, Uni¨ ersidad de Granada, 18071, Granada, Spain E-mail: [email protected] and [email protected] Communicated by Walter Feit Received August 7, 1997
We compute the number of elements of a minimal system of generators for the congruence of a numerical semigroup with embedding dimension minus multiplicity equal to zero, one, and two. For symmetric numerical semigroups we study the cases of embedding dimension minus multiplicity equal to one, two, and three. Q 1998 Academic Press
INTRODUCTION A numerical semigroup is a subset S of N closed under addition, containing the zero element and generating Z as a group. From this definition, one obtains that S has a minimal Žwith respect to the inclusion. system of generators as a semigroup, n 0 - n1 - ??? - n p 4 , verifying that the greatest common divisor of these generators is one. The number p q 1 is usually called the embedding dimension of S and the number n 0 is called the multiplicity of S. We define the semigroup homomorphism w : N pq 1 ª N as
w Ž a0 , a1 , . . . , a p . s a0 n 0 q a1 n1 q ??? qa p n p . Let us denote by s the kernel congruence of w . Thus, S ( N pq 1rs . Redei ´ shows in w3x that s is finitely generated. The scope of this paper is the computation of the number of elements of a system of generators for s with minimal cardinality. This problem has been already studied for some cases: Herzog studies in w2x this problem for the case p s 1 and p s 2; 567 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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Bresinsky treats in w1x the case when p s 3 and S is symmetric; Rosales studies in w5x the case p s n 0 y 1 and in w6x the case p s n 0 y 2 and S symmetric. The main purpose of this paper is the study of the cases p s n 0 y 2 and p s n 0 y 3 for S an arbitrary numerical semigroup and the cases p s n 0 y 3, and p s n 0 y 4 for S a symmetric numerical semigroup. In the first section, we obtain that if p s n 0 y 2, the number of elements in a system of generators for s with minimal cardinality is Ž n 0 y 1.Ž n 0 y 2.r2 y 1 or Ž n 0 y 1.Ž n 0 y 2.r2. We also determine the number of elements in a minimal presentation for S when p s n 0 y 3. In this case, we see that this number is between Ž n 0 y 2.Ž n 0 y 3.r2 y 2 and Ž n 0 y 2.Ž n 0 y 3.r2. As an application of the mentioned results, we give a classification of the semigroups with n 0 F 6 by the number of elements in a system of generators of its associated congruence with minimal cardinality. In the second section, we study the symmetric case. For p s n 0 y 2, the first author showed in w6x that ar s Ž n 0 y 1.Ž n 0 y 2.r2 y 1. We extend this result for the cases p s n 0 y 3 and p s n 0 y 4, and we show that the cardinality of a minimal system of generators for the congruence associated to the semigroup is Ž n 0 y 2.Ž n 0 y 3.r2 y 1 and Ž n 0 y 3.Ž n 0 y 4.r 2 y 1, respectively. As an application we study the cases in which S can be a complete intersection and we compute the number of a minimal system of generators for the congruence associated to a symmetric numerical semigroup with n 0 F 8. PRELIMINARIES In w4x, the first author gives an algorithm to compute a system of generators, r , for s with minimal cardinality. From the results given in that paper, it is derived that the concepts of a system of generators for s with minimal cardinality and a minimal system Žwith respect to the inclusion. of generators of s coincide. Next, we give a sketch of this construction. For every n g S, we define the graph G n s Ž Vn , En ., as Vn s n i g n 0 , . . . , n p 4 : n y n i g S 4 , En s w n i , n j x : n y Ž n i q n j . g S, i / j g 0, . . . , p 4 4 . We define rn as Ž1. If G n is not connected and G1n s Ž Vn1 , En1 ., . . . , G nr s Ž Vnr , Enr . are its connected components, then for every 1 F i F r we select an element
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a i s Ž a0 i , . . . , a p i . g N pq 1 _ 04 such that w Ž a i . s n and a k i s 0 for all n k i f Vni. Define rn s Ž a 1 , a 2 ., Ž a 1 , a 3 ., . . . , Ž a 1 , a r .4 . Ž2. If G n is connected, we define rn s B. The set r s Dng N rn is a system of generators for s with minimal cardinality Žsee w4x.. Given 0 / n g S, we define SŽ n. to be the Apery ´ set of n S Ž n . s s g S: s y n f S 4 . These sets are finite. In w4x, it is shown that if G n is not connected, then there exist s g SŽ n 0 . _ 04 and i g 1, . . . , p4 such that n s s q n i . The conductor of a numerical semigroup is the maximum integer not belonging to the semigroup. Such an element always exists, due to the fact that the group generated by a numerical semigroup is Z. It is also known that if x is the greatest element in S Ž n 0 . then the conductor, C, of S is C s x y n 0 . In w5x, the first author studies the relationship of a minimal set of generators for the congruence associated to a semigroup and a minimal set of generators for the congruence associated to the semigroup obtained adding its conductor to the given semigroup, provided that the conductor is greater than n 0 . ŽThe case C - n 0 only occurs when the semigroup is of the form ² n 0 , n 0 q 1, . . . , 2 n 0 y 1:. We may assume that the semigroup under study is not of this type, because this kind of semigroup can be studied using the results appearing in w5x.. Let us denote by S1 s S j C 4 . Then, S1 is generated by A1 s n 0 , . . . , n p , C 4 . The set A1 is a minimal system of generators for S1 if and only if x g SŽ n 0 . _ 0, n1 , . . . , n p 4 . If the set A1 is not a minimal system of generators for S1 , then it can be shown that ar s ar 1 , where r 1 is a minimal system of generators for the congruence associated to S1. Thus, if we are looking for bounds for ar , it does not matter if we study S1 instead of S. Let AX1 be a minimal system of generators for S1 and let S2 s S1 j C14 , with C1 the conductor of S1. Then, S ; S1 ; S2 and we can check once more if AX1 j C14 is a minimal generating system for S2 . If it is not so, we construct S3 . We continue this process until the semigroup constructed is minimally generated by the minimal system of generators plus the conductor of the previous semigroup. This procedure must stop, since it can be shown that there is a positive integer m such that Sm s S n 0 s ² n 0 , n 0 q 1, . . . , 2 n 0 y 1: s n 0 q N and in this case AXmy 1 j Cmy14 is a minimal system of generators for S m . Thus, we can assume that S s ² n 0 , . . . , n p : and S1 is minimally generated by n 0 , . . . , n p , C 4 . From the results presented by the first author in w5x, it is derived that r 1 has as many elements as r plus p q 2 elements Žcorresponding to the graphs G Cq n , . . . , G Cqn , G 2 C . minus the elements 0 p
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which arise when there is no path from n 0 to n i in the graphs of the form G xqn i , where x is the greatest element in SŽ n 0 . and i g 1, . . . , p4 . This idea will enable us to use the results known for semigroups with embedding dimension equal to the multiplicity of the semigroup in order to achieve bounds for the number of elements in minimal presentations of semigroups with embedding dimension their multiplicity minus one. Once this bound is obtained, we can use the same argument to compute a bound for semigroups with embedding dimension their multiplicity minus two. In each step, from the given semigroup we construct a new semigroup being under conditions already studied and then, we count how many elements n i verify that n i and n 0 belong to different connected components of G xqn i , where x is the greatest element in SŽ n 0 .. Finally, we recall the definition of two classes of semigroups. The semigroup S is symmetric if the set SŽ n. has a maximum with respect to the ordering induced by the addition in S. In this case, the semigroup ring K w S x s [s g S Kys is Gorenstein. The semigroup S is a complete intersection if ar is exactly p, which is the minimum possible number of elements that a system of generators for s may contain Žsee w2x..
1. NUMERICAL SEMIGROUPS WITH HIGH EMBEDDING DIMENSION For a numerical semigroup, the number of elements in SŽ n 0 . is exactly n 0 . Note that all the generators n i , i ) 0, are in S Ž n 0 .. Thus, the maximal embedding dimension for a semigroup is exactly n 0 Žfor the zero element is always in SŽ n 0 ... The semigroups having maximal embedding dimension have been studied by the first author in w5x. In the mentioned paper, these kinds of semigroups are called MED-semigroups, and it is shown that ar s n 0 Ž n 0 y 1.r2. An example of this kind of semigroups is the semigroups of the form S s ² n 0 , n 0 q 1, . . . , 2 n 0 y 1:, which already appeared in the preliminaries. 1.1. Numerical Semigroups with Embedding Dimension Their Multiplicity Minus One In this section, we study the minimal sets of generators for the congruences associated to numerical semigroups, S, such that S Ž n 0 . s 0, n1 , . . . , n p , w 4 , that is to say, numerical semigroups with embedding dimension p q 1 s n 0 y 1.
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Note that, since 0 / w g SŽ n 0 ., w can be written as w s a1 n1 q ??? qa p n p , with a i / 0 for some i. Thus, w y n i g SŽ n 0 .. Therefore w s n i q n j for some i, j g 1, . . . , p4 Žnot necessarily different.. Let S1 s ² n 0 , . . . , n p , C :. As we have seen in the preliminaries, we can assume that S1 is minimally generated by n 0 , . . . , n p , C 4 . Note that this is equivalent to the fact that w is the greatest element in SŽ n 0 ., because if n i is the greatest element in SŽ n 0 . for some i, then n i s Ž n i y n 0 . q n 0 s C q n 0 and therefore n 0 , . . . , n p , C 4 does not minimally generate S1. Hence, S1 is a MED-semigroup and ar 1 s n 0 Ž n 0 y 1.r2. In order to count the elements belonging to r , we must count how many generators n i , 1 F i F p verify that n i and n 0 are in different connected components of G wqn i . Let w s n i q n j . If n k and n 0 are not in the same connected component of G wq n k for some k g 1, . . . , p4 , then w q n k y Ž n i q n 0 . f S and w q n k y Ž n j q n 0 . f S. This implies that n j q n k , n i q n k g SŽ n 0 . which leads to n i s n j s n k . Thus, the only graph that can add a new element to r is G wq n i when w s 2 n i and w cannot be expressed in another way. Let us show that, as a matter of fact, in this case n i and n 0 are in different connected components of G wq n i . If it were not so, there would exist a path from n i to n 0 , which would mean that w q n i y Ž n i q n k . g S for some k / i. Nevertheless, w q n i y Ž n i q n k . s 2 n i y n k s w y n k and if w y n k g S, then w must admit an expression different from 2 n i , a contradiction. We get the following theorem: THEOREM 1. Let S s ² n 0 , . . . , n p : be a numerical semigroup such that SŽ n 0 . s 0, n1 , . . . , n p , w4 and let r be a minimal system of generators for the congruence associated to S. Ž1.
If w s n i q n j for some i, j g 1, . . . , p4 , i / j then ar s
Ž2.
Ž n0 y 1. Ž n0 y 2. 2
y 1.
Otherwise, ar s
Ž n0 y 1. Ž n0 y 2. 2
.
Proof. Let q be the number of elements in r arising from the fact that n i is not in the same connected component of n 0 in G wqn i . We have just shown that if w s n i q n j with i / j, then q s 0. Otherwise, q s 1. This concludes the proof, because ar 1 s n 0 Ž n 0 y 1.r2 s ar q Ž p q 2. y q s ar q n 0 y q and therefore ar s Ž n 0 y 1.Ž n 0 y 2.r2 y 1 q q.
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As an application of the previous theorem, one can get a classification of semigroups having n 0 F 5 by the number of elements in a minimal system of generators for its congruence. The only cases which are not complete intersections and are not covered by w2x or w5x are n 0 s 4, p s 2 and n 0 s 5, p s 3. These are covered by Theorem 1. 1.2. Numerical Semigroups with Embedding Dimension Their Multiplicity Minus Two In this case, S Ž n 0 . s 0, n1 , . . . , n p , w 1 , w 2 4 . As in the previous section, we can assume that S s ² n 0 , . . . , n p : and that S1 is minimally generated by n 0 , . . . , n p , C 4 , that is to say, the greatest element in S Ž n 0 . is in w 1 , w 2 4 . We will suppose that w 2 is such a maximum. Hence, S1 s ² n 0 , . . . , n p , C : verifies that S1Ž n 0 . s 0, n1 , . . . , n p , C, w 14 and therefore ar 1 g
Ž n0 y 1. Ž n0 y 2. 2
y 1,
Ž n0 y 1. Ž n0 y 2. 2
.
Thus, we are interested in the elements n i such that n 0 and n i are in different connected components of G w 2qn i . Let q be the number of such elements. We have ar s ar 1 y n 0 q 1 q q. The following cases are possible: Case 1. aVw 2 G 3. Take n i , n j , n k as three elements in Vw 2 . If n t and n 0 are in a different connected component of G w 2qn t then n i q n t , n j q n t , n k q n t g SŽ n 0 . _ 0, n1 , . . . , n p 4 Žif, for instance, n i q n t y n 0 g S, take n s g Vw 2yn i ; then w 2 q n t y Ž n s q n 0 . s Ž w 2 y n i y n s . q Ž n i q n t y n 0 . g S .. But this is impossible, because aSŽ n 0 . _ 0, n1 , . . . , n p 4 s 2. Thus, in this case q s 0 and Ža. If aVw G 2 then 1 ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 2.
Žb. If aVw s 1 then 1 ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
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Case 2. aVw 2 s 2. Let Vw 2 s n i , n j 4 . From the fact that aSŽ n 0 . _ 0, n1 , . . . , n p 4 s 2 it is easily derived that w 2 must be equal to n i q n j . If n k is such that there is no path from n 0 to n k in G w 2qn k, then n i q n k , n j q n k g SŽ n 0 .. Hence, n i s n k and w 1 s 2 n i , or n j s n k and w 1 s 2 n j . Let us study the different possibilities for aVw 1. Ža. aVw s 1. ŽNote that in this case w 1 s 2 n t for some t and this 1 expression is unique.. Ži. If w 1 / 2 n i and w 1 / 2 n j then n k and n 0 are in the same connected component of G w 2qn k, for all k. Hence, q s 0 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
Žii. If w 1 s 2 n i then this expression is unique. Hence, w 2 q n i y Ž n i q n k . g S if and only if k g i, j4 and w 2 q n i y Ž n j q n k . g S if and only if k s i, which means that the connected component of n i in G w 2qn i is n i , n j 4 . Thus, q s 1 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
.
Žiii. If w 1 s 2 n j we obtain the same result. Žb. aVw G 2. 1 Ži. If w 1 / 2 n i and w 1 / 2 n j then n k and n 0 are in the same connected component of G w 2qn k, for all k. Hence, q s 0 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 2.
Žii. If w 1 s 2 n i then, since aVw G 2, w 1 can be expressed as 1 n k q n l with i f k, l 4 . Let us show that n i and n 0 are in the same connected component of G w 2qn i . Note that w 2 q n i y Ž n j q n k . s 2 n i y n k s n l g S and that w 2 q n i y Ž n j q n l . s 2 n i y n l s n k g S. Thus, n i , n j , n k , n l 4 are in the same connected component of G w qn . Since 2 i i f k, l 4 then n j q n k and n j q n l are different from w 2 . This means that n j q n l f SŽ n 0 . or n j q n k f SŽ n 0 . Žor both.. In the first case, w 2 q n i y Ž n k q n 0 . s n j q n l y n 0 g S and in the second, w 2 q n i y Ž n l q n 0 . s n j q n k y n 0 g S, which implies in both cases that n 0 is connected with n i . Thus, in this case q s 0 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 s
Ž n0 y 2. Ž n0 y 3.
Žiii. If w 1 s 2 n j we get the same result.
2
y 2.
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Case 3. aVw 2 s 1. Then, w 2 s 2 n i or w 2 s 3n i for an element i g 1, . . . , p4 . In both cases, the expression of w 2 is unique and therefore n i is an isolated point of G w 2qn i Žotherwise, w 2 q n i y Ž n i q n k . s w 2 y n k g S for some k / i which is impossible, because aVw 2 s 1.. Thus, q G 1. Ža. aVw s 1. If n k and n 0 are in different connected components of 1 G w 2qn k, then n i q n k g SŽ n 0 ., and this only occurs if i s k. Hence, q s 1 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y n0 q 1 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
.
Žb. aVw s 2. If w 2 s 3n i then w 1 must be 2 n i and this expression 1 must be unique, since aVw 2 s 1. But this implies that aVw 1 s 1, which is a contradiction. Then, w 2 s 2 n i , w 1 s n k q n l , for some k / l, and these expressions are unique. If n t Ž t / i . and n 0 are not connected in G w 2qn t then w 2 q n t y Ž n i q n 0 . f S, which means that n i q n t g SŽ n 0 . and therefore i, t 4 s l, k 4 . Hence, there is no s f i, t 4 such that n s and n 0 are not connected in G w 2qn s. Let us show that if w 1 s n i q n t then n i , n t 4 is a connected component of G w 2qn t and therefore q s 2. The element w 2 q n t y Ž n i q n s . s w 1 y n s belongs to S if and only if s g i, t 4 , because w 1 is expressed in an unique way. Besides, w 2 q n t y Ž n t q n s . s w 2 y n s is in S only if s s i, because Vw 2 s n i 4 . Thus, ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 q 2 s
Ž n0 y 2. Ž n0 y 3. 2
.
Žc. aVw G 3. As before, w 2 must be equal to 2 n i . If n j Ž j / i . and 1 n 0 are in different connected components of G w 2qn j , then w 2 q n j y Ž n i q n 0 . f S and therefore n i q n j s w 1. Since aVw 1 G 3, we can choose n k g Vw 1 _ n i , n j 4 . Note that w 2 q n j s 2 n i q n j s w 1 q n i and consequently n i , n j , and n k are in the same connected component of G w 2qn j . Let us show that n k and n 0 are connected. The element w 2 q n j y Ž n k q n 0 . s w 1 q n i y Ž n k q n 0 . s Ž w 1 y n k . q n i y n 0 belongs to S if and only if w 1 y n k q n i f SŽ n 0 . Žnote that w 1 y n k g S .. It is enough to show that w 1 y n k q n i / w 1 and w 1 y n k q n i / w 2 . Since n i / n k , w 1 y n k q n i cannot be w 1. If w 1 y n k q n i s w 2 , then w 1 q n i s 2 n i q n k and therefore n i q n j s w 1 s n i q n k , which means that n j s n k , and this is not possible. Thus, q s 1 and ar s
Ž n0 y 1. Ž n0 y 2. 2
y 1 y n0 q 1 q 1 s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
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We have shown the following theorem: THEOREM 2. Let S s ² n 0 , . . . , n p : be a numerical semigroup such that SŽ n 0 . s 0, n1 , . . . , n p , w 1 , w 2 4 and let r be a minimal system of generators for the congruence associated to S. Then ar g s
p Ž p q 1. 2
y 2,
p Ž p q 1.
Ž n0 y 2. Ž n0 y 3. 2
2 y 2,
Ž n0 y 2. Ž n0 y 3. 2
.
Furthermore, if we assume w 1 - w 2 , then ar s Ž n 0 y 2.Ž n 0 y 3.r2 y 2 if and only if aVw 2 G 2 and aVw 1 G 2. ar s Ž n 0 y 2.Ž n 0 y 3.r2 y 1 if and only if aVw 2 G 3 and aVw 1 s 1, or aVw 2 s 2, aVw 1 s 1, and Vw 2 l Vw 1 s B, or aVw 2 s 1 and aVw 1 G 3. ar s Ž n 0 y 2.Ž n 0 y 3.r2 if and only if aVw 2 s 2 and Vw 1 ; Vw 2 , or aVw 2 s 1 and aVw 1 F 2. v v
v
As we did in the previous section for n 0 g 1, . . . , 54 , one can now study the case n 0 s 6 and see which are the possible values of ar . The cases that are not complete intersections and do not lay in the scope of the results in w2x nor in the results appearing in w5x are p s 3 and p s 4. The former case is covered by Theorem 2 and the latter by Theorem 1. 1.3. Numerical Semigroups with Embedding Dimension Their Multiplicity Minus Three From the results obtained in the previous sections one could think that the bounds for this case are Ž n 0 y 3.Ž n 0 y 4.r2 y 3 F ar F Ž n 0 y 3.Ž n 0 y 4.r2. If we take S s ²7, 8, 10, 19:, we get SŽ7. s 0, 8, 10, 16, 18, 19, 204 and ar s 7 ) Ž7 y 3.Ž7 y 4.r2, and therefore these bounds are not correct. If our computations are correct, the number of elements in r is in the interval wŽ n 0 y 3.Ž n 0 y 4.r2 y 3, Ž n 0 y 3.Ž n 0 y 4.r2 q 1x. We do not include the proof here because of the big amount of cases and subcases arising. This ‘‘bad’’ result makes one think of alternative ways of study for numerical semigroups with not so high embedding dimension. 2. SYMMETRIC NUMERICAL SEMIGROUPS WITH HIGH EMBEDDING DIMENSION If S is a symmetric semigroup then the set S Ž n 0 . has a maximum with respect to the ordering induced in S by the addition, and this maximum cannot belong to a minimal system of generators for S, but for n 0 s 2.
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This implies that the maximal embedding dimension is exactly n 0 y 1. In w6x, the first author denotes this kind of semigroups by MEDSY-semigroups and shows that ar s Ž n 0 y 1.Ž n 0 y 2.r2 y 1, which coincides with the first case of Theorem 1, but for n 0 equal to two or three. Note that symmetric numerical semigroups always lie in the first case of Theorem 1 because if SŽ n 0 . s 0, n1 , . . . , n p , w4 has a maximum, then w must be this maximum and it must be of the form w s n i q n j for some i / j Žprovided p G 2.. If S is a complete intersection, then S is Gorenstein and therefore must lie in the first case. Under this setting, n 0 must be equal to 4 or equal to 1 Ž S s N.. Thus, for n 0 G 5 there are no complete intersection semigroups with p s n 0 y 2. If p s n 0 y 3, and SŽ n 0 . s 0, n1 , . . . , n p , w 1 , w 2 4 then w 2 , the maximum of SŽ n 0 ., must be w 1 q n i for some n i , which leads to w 1 s 2 n i Žif aVw 1 G 2 then take n k / n l g Vw 1; the elements n k q n i , n l q n i g SŽ n 0 . _ w 2 4 , which means that w 1 s n k q n i s n l q n i , a contradiction.. Since w 2 y n j g S for all j, we have that the expression w 2 s 3n i is not unique, and therefore if p G 3 Žthe cases p s 0 and p s 1 are trivial and the case p s 2 has been already studied by Herzog in w2x., then aVw 2 G 3 which means that ar s
Ž n0 y 2. Ž n0 y 3. 2
y 1.
Note that if S is a complete intersection and p s n 0 y 3, then from the equation ar s pŽ p q 1.r2 y 1 s p we get n 0 s 5. Thus, if n 0 G 6, there are no complete intersection numerical semigroups with their multiplicity equal to their embedding dimension minus two. Finally, let us study the case p s n 0 y 4: S Ž n 0 . s 0, n1 , . . . , n p , w 1 , w 2 , w 3 4 . The following theorem provides a result analogous to the ones obtained so far for p ) n 0 y 4. THEOREM 3. Let S s ² n 0 , . . . , n p : be a symmetric numerical semigroup such that SŽ n 0 . s 0, n1 , . . . , n p , w 1 , w 2 , w 2 4 , p G 3, and let r be a minimal system of generators for the congruence associated to S. Then ar s
Ž n0 y 3. Ž n0 y 4. 2
y 1.
Proof. Note that if S is symmetric then the greatest element x in SŽ n 0 . is in w 1 , w 2 , w 34 , because x y y g S for all y g SŽ n 0 .. Without loss of generality we can assume that x s w 3 and that w 1 - w 2 . Thus, S1 s S j
ON NUMERICAL SEMIGROUPS
577
C 4 is minimally generated by n 0 , . . . , n p , C 4 and S1 is under the conditions of the previous section. As we have done before, we must check how many elements in r arise from the fact that n i and n 0 are in different connected components of G w 3qn i for n i g 1, . . . , p4 . Assume that n i , i G 1, is such that n i and n 0 are in different connected components of G w 3qn i . Then, since w 3 y n k g S for all k g 1, . . . , p4 we get that if n 0 is not connected with n i in G w 3qn i , then the elements n k q n i are in SŽ n 0 . for all k g 1, . . . , p4 Žotherwise the edges w n i , n k x and w n k , n 0 x are in EŽG w 3qn i ... But p G 3 and n1 q n i , . . . , n p q n i are all different and non-comparable in S. This is a contradiction with the fact that the maximum number of non-comparable elements in S Ž n 0 . _ 0, n1 , . . . , n p 4 is two. This means that ar s ar 1 y n 0 q 2. If we want to show that ar s Ž n 0 y 3.Ž n 0 y 4.r2 y 1, then we must prove that ar 1 s Ž n 0 y 2.Ž n 0 y 3.r2 and this is equivalent to demonstrating Žby Theorem 2. that aVw 2 s 2 and Vw 1 ; Vw 2 , or that aVw 2 s 1 and aVw 1 F 2. First of all, let us show that aVw 2 F 2 and that aVw 1 F 2. If n i , n j , n k are three different elements in Vw 2 then take n l g Vw 3yw 2 . The elements n l q n i , n l q n j , n l q n k are three different non-comparable Žw.r.t. the ordering induced in S by the addition. elements in S Ž n 0 . _ 0, n1 , . . . , n p 4 , and this is not possible. In the same way, it is shown that aVw 1 F 2. Observe that w 3 y w 2 g SŽ n 0 ., and therefore w 3 y w 2 s w 2 , w 3 y w 2 s w 1 or w 3 y w 2 g n1 , . . . , n p 4 . If w 3 y w 2 s w 2 , then take n k g Vw 2 . The element w 2 q n k g SŽ n 0 . is greater than w 1 and it is neither w 2 nor w 3 , a contradiction. Analogously, we can show that w 3 y w 2 / w 1. Thus, w 3 y w 2 g n1 , . . . , n k 4 . We have seen that w 3 y w 1 / w 2 and therefore w 3 y w 1 s w 1 or w 3 y w 1 g n1 , . . . , n p 4 . If w 3 y w 1 s w 1 , take n k g Vw 1. Then, w 1 q n k g SŽ n 0 . _ w 3 , w 1 4 and therefore w 1 q n k s w 2 . Briefing, we get that there are two possible cases: Ž1. There exists i / j such that w 3 s w 2 q n i s w 1 q n j . Ž2. There exists i, j such that w 3 s w 2 q n i and w 2 s w 1 q n j . Let us study each case separately: Ž1. It is enough to show that if aVw s 2 then Vw ; Vw , because if 2 1 2 aVw 2 s 1 then, since aVw 1 F 2, we are done. Assume that aVw 2 s 2. Then w 2 s n s q n t for some s - t. As w 3 s w 2 q n i , then n s q n i , n t q n i g SŽ n 0 . _ w 34 . Hence, n s q n i s w 1 and n t q n i s w 2 s n s q n t , which means that n s s n i and consequently w 1 s 2 n i . Besides, w 3 s w 1 q n j s
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2 n i q n j s w 2 q n i and this leads to w 2 s n i q n j Ž i, j4 s s, t 4.. Let us assume that w 1 can be expressed in a different way. Then w 1 s n k q n l with i f k, l 4 and k / l. Since w 3 s w 1 q n j , n k q n j , n l q n j g SŽ n 0 .. However, the expression w 2 s n i q n j is unique, which implies that w 1 s n k q n j s n l q n j , and this is impossible since k / l. We have shown that Vw 1 s n i 4 ; n i , n j 4 s Vw 2 . Ž2. If we show that aVw s 1 then, since aVw F 2, we are done. If 2 1 aVw 2 s 2, then w 2 s n k q n l , for some n k - n l . Hence, n k q n i , n l q n i g SŽ n 0 . _ w 34 , because w 3 s w 2 q n i . This implies that n k q n i s w 1 and n l q n i s w 2 , but this leads to n l q n i s w 2 s w 1 q n j s n k q n i q n j , which means that n l s n k q n j , and this is impossible. The restriction p G 3 is unimportant, because for the cases p s 0 and p s 1 the semigroup is a complete intersection, and the case p s 2 has been already studied by Herzog for S a numerical semigroup not necessarily symmetric. With the results obtained in this paper one can compute the possible values of ar for the symmetric numerical semigroups with n 0 F 8. Observe that if n 0 F 6, the study can be done using the results exposed in the previous sections together with the results appearing in w2, 5x. Recall that if n 0 G 3 and the semigroup is symmetric, then p F n 0 y 2. Note also that, due to the results obtained by Herzog in w2x, if p s 2 then the semigroup is symmetric if and only if it is a complete intersection. Thus, for n 0 s 7 and p F 2 we get complete intersections; for p s 4 we can apply Theorem 3; for p s 5 we get a MEDSY-semigroup. Finally, if n 0 s 8 and p F 2 we get once more a complete intersection semigroup; for p g 4, 5, 64 we can use the results exposed in this paper; for p s 3, using the results obtained by Bresinsky in w1x, we get that ar g 3, 54 . Note that for n 0 s 8 and p s 3 s n 0 y 5 the statement ar s Ž n 0 y 4.Ž n 0 y 5.r2 y 1 is false. Taking, for example, S s ²8, 10, 12, 15:, one can compute r and obtain that ar s 3 / 4 = 3r2 y 1 s 5. REFERENCES 1. H. Bresinsky, Symmetric semigroups of integers generated by 4 elements, Manuscripta Math. 17 Ž1975., 206]219. 2. J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 Ž1970., 175]193. 3. L. Redei, ‘‘The Theory of Finitely Generated Commutative Semigroups,’’ Pergamon, ´ Elmsford, NY, 1965. 4. J. C. Rosales, An algorithmic method to compute a minimal relation for any numerical semigroup, Internat. J. Algebra Comput. 6, No. 4 Ž1996., 441]455. 5. J. C. Rosales, On numerical semigroups, Semigroup Forum 52 Ž1996., 307]318. 6. J. C. Rosales, On symmetric numerical semigroups, J. Algebra 182 Ž1996., 422]434.