Every positive integer is the Frobenius number of an irreducible numerical semigroup with at most four generators

(1)

Ark. Mat., 42 (2004), 301-306

Every positive integer is the Frobenius number of an irreducible numerical semigroup with at most four generators

P e d r o A. Garcfa-Ss a n d Jos~ C. Rosales(~)

A b s t r a c t . Let g be a positive integer. We prove that there are positive integers nl, n2, na and n4 such that the semigroup S = ( n l , n 2 , n3,n4} is an irreducible (symmetric or pseudo- symmetric) numerical semigroup with g(S)--g.

I n t r o d u c t i o n

Let n l , ..., np be positive integers with g c d { n l , ..., r i p } = 1 (where as usual gcd s t a n d s for greatest c o m m o n divisor). T h e n it is not h a r d to show t h a t there are finitely m a n y elements n t h a t c a n n o t be expressed as n = a l n l + . . . + a p n p for some n o n n e g a t i v e integers a l , ..., ap. T r a n s l a t e d to numerical semigroups, this is equiva- lent to say t h a t if we consider t h e numerical s e m i g r o u p S g e n e r a t e d by { h i , . . . , rip}, t h a t is, S - (hi, ..., np} = {al rtl +... + avn p l a l , . . . , ap C N } , t h e n the set N \ S is finite (where N denotes t h e set of n o n n e g a t i v e integers). T h e m a x i n m m of this set is usually k n o w n as t h e Frobenius number of S a n d it is here d e n o t e d by g(S). T h e p r o b l e m of d e t e m f i n i n g a general formula for g(S) in terms of h i , ..., np is k n o w n as t h e Frobenius problem, which goes back to [11], where an explicit formula for p = 2 is given ( g ( ( n l , ' n ~ } ) = n l n 2 - n l - n 2 ) . It can be shown (see [3]) t h a t no general formula of a certain t y p e can be found even tbr t h e case p = 3 . A nice survey on t h e state of t h e a r t of t h e Frobenius p r o b l e m can be found in [5].

For a given positive integer g, t h e s e m i g r o u p S = { g + l , g + 2 , . . . , 2 g - 1 ) = {0, g + l , - + } (here t h e s y m b o l --+ is used to indicate t h a t every h E N w i t h n > g + l belongs to the set) fulfills the trivial condition g(S) = 9 . D e n o t e t h e set of numerical semigroups with Frobenius n u m b e r 9 by $ ( g ) . In a recent p a p e r [9] t h e a u t h o r s have shown t h a t for every positive integer g, there always exist n l , n2 a n d n3 such

(1) This work has been supported by the project BFM2000-1469.

(2)

302 Pedro A. Garcla-Ss and Jos6 C. Rosales

t h a t (nl, n2, n 3 ) ~ $ ( 9 ) (or in other words, g ( ( n l , n2, n a } ) - g ) . Among the elements in $(9), there are some numerical semigroups t h a t have some relevance in ring the- ory, since their associated semigroup rings are Gorenstein and Kunz. These are the so called symmetric and pseudo-symmetric mtmerical semigroups, respectively, and have been characterized in m a n y ways (see for instance [2] and [4]). One of these characterizations states t h a t they are precisely those numerical semigroups in

$(g) t h a t are m a x i m a l with respect to set inclusion. B o t h concepts (symmetric and pseudo-symmetric) can be unified into the single concept of irreducible numerical semigroups (see for instance [7] and [8]). A numerical semigroup is irreducible if it cannot be expressed as the intersection of two numerical semigroups properly containing it. If S is a numerical semigroup with g(S) odd (respectively even), then S is symmetric (respectively pseudo-symmetric) if and only if S is irreducible.

In this p a p e r we give an easy procedure to find, for any fixed positive integer 9, an irreducible numerical semigroup with at most four generators and having g as Frobenius number. Moreover, four is the least n u m b e r of generators needed for the general case, even though in some cases an irreducible numerical semigroup with three (or two if g is odd) generators can be found.

Acknowledgement. T h e authors would like to t h a n k the referee for her/his com- ments and suggestions.

1. P r e l i m i n a r i e s

Let S be a numerical semigroup. We say t h a t {nl, ... , n p } C S is a system of generators of S if S = ( n l , ... ,rip}. For n E S \ { 0 } we define the Ap~ry set of n in S (see [1]) as tile set

A p ( S , n ) - { s E S I s - n • S } .

It is not difficult to prove t h a t this set has exactly n elements, which are w 0 = 0, w,, ..., w~ j, where wi is the least element in S congruent with i modulo n. From the definition of Ap(S, n) it also follows t h a t if x=y+zCAp(S, n) with y, zES, then b o t h y and z are again in Ap(S, n). This idea is implicitly used several times in the rest of the paper. Besides, it is not hard to show t h a t g ( S ) + n is the greatest element in Ap(S, n).

T h e cardinality of the set

T ( S ) = {x E Z \ S l x + s ~ S for all 8 E S \ { 0 } }

is known as the t y p e of S (see for instance [2] and [4]; as usual Z denotes the set of integers). S y m m e t r i c numerical semigroups are those numerical semigroups of

(3)

Positive integers as Probenius numbers of irreducibles 303

type 1 (this forces T ( S ) - { g ( S ) } , since by the definition of the Frobenius number of S, for every h E N \ { 0 } , the element g+n belongs to S). Pseudo-symmetric numerical semigroups are characterized by T(S) { 89 (see [2] and [4]).

For a given numerical semigroup one can define the order relation _<s as follows:

for cc,~IES, x<_sy holds if y - z ~ S . In [6] it is shown that for every n C S \ { 0 } , we have that T ( S ) = { x - n l x ~ m a x < s Ap(S, n)} (max<~ stands for maximal elements with respect to <_s). Hence from [4], we deduce that, with n E S \ { 0 } fixed, S is irreducible if and only if

{ { g ( S ) + n } , if g(S) is odd, max_< s A p ( S , n ) = {89 g(S)+n}, if g ( S ) i s even.

2. M a i n r e s u l t

L e m m a 1. Let g be a positive integer. If 2{g, then S - (2, g + 2)

is an irreducible numerical sernigroup with g ( S ) = g . The proof is trivial.

L e m m a 2. Let g be a positive integer. If 2]g and 3~ then S = (3, 1g+3, g+3}

is an irreducible numerical sernigroup with g ( S ) = g .

Proof. First note that gcd{3, 8 9 whence S is a numerical semigroup. Also, Ap(S, 3)={O, 19+3,g+3} and thus max<_sAp(S, 3)={ 89 g+3 }.

This implies that g ( S ) - g + 3 3 g and that S is irreducible. []

L e m m a 3. Let g be a positive integer. If 21g and 4~9, then S=(4, 89

is an irreducible numerical semigronp with g ( S ) = g .

Proof.

Since

21g

and 4~.q, we have gcd{4, ~ }=1. Hence lg gcd{4, 89 89 = 1 and S is a numerical semigroup. We prove that

(1) Ap(S, 4) = {0, 89 89 9 + 4 } .

(4)

304 P e d r o A. G a r c f a - S A n c h e z a n d Jos~ C. R o s a l e s

But this can easily be deduced from the following two facts:

(a) 89 and 89 are minimal in S with odd remainder modulo 4, and ^{a r e} clearly in different classes;

(b) g + 4 is minimal in S with even (and nonzero) remainder modulo 4.

Since (1) holds, we conclude that g ( S ) = 9 and T ( S ) = { } 9 , 9 } . []

L e m m a 4. Let g be a positive integer. Assume that 21g, 3lg and 41g. Let (~

and q be such that g=3~q, with (3, q ) - 1 (and thus 4[@. Then s : <3 ~ 1 89 3, 3 ~ ~ + ~ + 3, 3 ~ 89 89 is an irreducible numerical semigroup with g(S) g.

, 1 1

Proof

Let ~a=3 ~+1 ~2=89 ~3=3~1q+89 and ~ 4 = 3 ~ q + ~ q + 3 . As (3, q ) = l , we have gcd{n~,n2,n3,n4} 1. The reader can check that the following equalities hold:

(i) 2n4=~qnl+n2+n3;

(ii)

2n3--n2+n4;

(iii) n 3 @ n 4 = ~qnl +2n2; 1

(iv) (ln~ + 1),~2 =,~1 +'~4;

(vi) (ln~ 1)n~+~ (88

Every element in Ap(S, n~) is of the form an2 +bna +cn4. By (repeatedly) using (i), (ii) and (iii) we can assuine that one of the following three cases holds:

(a) b = c = 0 ; (b) b=0 and c=1;

(e) b = l and e=0.

1 9c~ In the case (b), it ibllows from (v) It follows from (iv) that a is always < 5r~1 = o .

that a < ~ n l , and in the case (c), a must be less than 8 9 in view of (vi). Since

# Ap(S, n l ) = n l , one can easily deduce that

Ap(S, nl)={O,r~2,2n2, 9 .. , 5rt17~2, n 3 , r t 3 - o n 2 , . . - , ~ z 3 + ( S r t l -- 2 ) ~t2, 1 1

'/~4, 7/~4 ~ - n 2 , ... , n4-~- ( l n l -- 1) 7~2 }.

Now we prove that m a x < s A p ( S , n l ) = { l n l n 2 , ( s n l - 1 ) n 2 + r ~ 4 } . 1 From the shape of Ap(S, nl) it is clear that

m a x < s A p ( S , n l ) C { ~ r t l n 2 , T L 3 + ( ~ T L 1 - - 2 ) r t 2 , n 4 - } - ( l n l 1 1)n2}.

After some simplifications one gets that

( } n l - 1 ) n 2 + n 4 (r~3+(~n~ 2 ) r t 2 ) = n 4 - ~ 3 + n ~ ,

(5)

Positive integers as Frobenius numbers of irreducibles 305

which in view of (ii) is equal to ha, which trivially belongs to S. This implies t h a t na + ( 5 n l - 2) ~max_< s Ap(S, 7~1). Since l n s n 2 c A p ( & hi), we have t h a t 1

~Tt17t2 - ~ t l = 3 a 1 ~ q ~ S . Thus ~'~4@(1,t1--1)~%2-- 17~ 7% __0c~1~4 5 1 2 o ~ S . Hence

max<_s. A p ( S , n z ) = { ~ n z r t 2 , ( l n l 1 ) n 2 + n 4 } .

This in particular implies t h a t g(S) (lanl 1 ) r ~ 2 + n n - n l = 3 ~ q and t h a t T ( S ) = { 8 9 Therefore S is irreducible. []

T h e o r e m 5. Let g be a positive integer. Then there exist nl,n2,r~a,r~4EN such that S = <nl , n2, na, nn} is an irreducible numerical semigroup with g ( S ) = g .

Proof. If g is not a multiple of 2, then by L e m m a 1 we can choose n 1 = 2 and rt2=rt3 = r ~ 4 = g + 2 . If on the contrary g is a multiple of 2, then we distinguish two cases.

(1) I f g is not divisible by 3, t h e n in view of Leinma 2, for n 1 = 3 , n 2 = 8 9 and

~ t a = f t 4 = g + 3 , the semigroup S=<nl, T~2, Tt3, n4} is an irreducible numerical semigroup with g ( S ) = g .

(2) If g is divisible by 3, then

(a) either g is not divisible by 4, and thus we can apply L e m m a 3 and take

nz 4, n 2 = 8 9 n4 89

(b) or g is divisible by 4, and in this case we get the desired semigroup by using L e m m a 4. []

If one computes the set of irreducible numerical semigroups with Frobenius n u m b e r 12 as explained in [10], one gets t h a t this set contains only two elements:

<5, 8, 9, 11} and <7, 8, 9, 10, 11,

la>.

This implies t h a t the bound of four generators obtained in T h e o r e m 5 cannot be improved.

R e f e r e n c e s

1. APIARY, R., Sur les branches superlin6aires des courbes alg6briques, C. R. Acad. Sci.

Paris 222 (1946), 1198 1200.

2. BARUCCI, V., DOBBS, D. E. and FONTANA, M., Mazirnality Properties in Numeri- cal Sernigro~aps and Applications to Or~e-Dimensional Analytically Irreducible Local Domains, Mere. Amer. Math. Soc. 125, Amer. Math. Soc., Providence, R. I., 1997.

3. CURTIS, F.~ On formulas for the Frobenius number of a numerical sernigroup, Math.

Scan& 67 (1990), 190 192.

(6)

306 Pedro A. Garcfa-S~nchez and Josd C. Rosales:

Positive integers as Frobenius numbers of irreducibles

4. FROBERG, i:{., GOTTLIEB, C. and HAGGKVIST, R., On numerical semigroups, Semi- group Forum 35 (1987), 63 83.

5. RAMfREZ ALFONSfN, J. L., The Diophantine Frobenius problem Preprint, For- schungsinstitut fiir Diskrete Mathematik, Bonn, 2000.

6. ROSALES, J. C. and BRANCO, M. B., Numerical semigroups t h a t can be expressed as an intersection of symmetric numerical semigroups, J. Pure Appl. Algebra 171 (2002), 303 314.

7. ROSALES, J. C. and BRANCO, M. B., Decomposition of a numerical semigroup as an intersection of irreducible numerical semigroups, Bull. Belg. Math. Soc. 9 (2002), 373 381.

8. ROSALES, J. C. and BRANCO, M. B., Irreducible numerical semigroups, Pacific J.

Math. 209 (2003), 131-143.

9. ROSALES, J. C., GARCIA-SANCHEZ, P. A. and GARCfA-GARC1A, J. I., Every positive integer is the Probenius number of a numerical semigroup with three generators, Preprint, 2003.

10. ROSALES, J. C., GARCfA-S/~NCHEZ, P. A., GARCfA-GARCfA, J. I. and JIM~NEZ- MADRID, J. A., The oversemigroups of a numerical semigroup, Semigrvup Forum 67 (2003), 145 158.

11. SYLVESTER, J. J., Mathematical questions with their solutions, Educational Times 41 (1884), 21.

Received February 11, 2003 Pedro A. Garcfa-Sgnchez

D e p a r t a m e n t o de Algebra Universidad de G r a n a d a ES-18071 G r a n a d a Spain

emaih pedro@ugr.es Jos6 C. Rosales

D e p a r t a m e n t o de Algebra Universidad de G r a n a d a ES-18071 G r a n a d a Spain

email: jrosales@ugr.es