Quotient Numerical Semigroups (work in progress)

Transcription

1 Quotient Numerical Semigroups (work in progress) Vítor Hugo Fernandes FCT-UNL/CAUL (joint work with Manuel Delgado) February 5, 2010 Iberian meeting on numerical semigroups Granada (Universidad de Granada) 1 / 16

2 Definition Let N be a numerical semigroup and let I = {n N + n > F (N)} be the canonical ideal of N. The quotient numerical semigroup (qns) associated to N is the Rees quotient Q(N) = N + /I, with N + = N \ {0}. Recall that N + /I = {{x} x N + \ I } {I } and so we may identify Q(N) with the semigroup (N + \ I { }, ), where the binary operation is defined by { x + y if x + y < F (N) x y = otherwise, and x = x = =, for all x, y N + \ I (usually we denote simply by +). Notice that, with this identification, a qns may be associated with several distinct numerical semigroups (Universidad de Granada) 2 / 16

3 Examples N =< 1 > = Q(N) = { }, I = N + ( the trivial qns) N =< 3, 5 > = Q(N) = {3, 5, 6, }, I = {8, } (Universidad de Granada) 3 / 16

4 N =< 4, 5 > = Q(N) = {4, 5, 8, 9, 10, }, I = {12, } (Universidad de Granada) 4 / 16

5 N =< 4, 7 > = Q(N) = {4, 7, 8, 11, 12, 14, 15, 16, }, I = {18, } (Universidad de Granada) 5 / 16

6 N =< 7, 8, 9 > = Q(N) = {7, 8, 9, 14, 15, 16, 17, 18, }, I = {21, } N =< 7, 8, 9, 20 > = Q(N) = {7, 8, 9, 14, 15, 16, 17, 18, }, I = {20, } (Universidad de Granada) 6 / 16

7 Let S = {a 1 < a 2 < < a n < } be a qns. Notice that, by considering the natural order induced by the usual order of N, we may view S as an (linearly) ordered semigroup. Let A be the set of irreducible/indecomposable elements of S, i.e. A = S \ (S + S). Then A is the unique minimal generating set (mgs) of S. Observe that a qns may not be completely defined by its mgs and addition in N: in general, we also need to know, for instance, its largest finite element (i.e. a n ) or the number of finite elements (i.e. n). Examples S = {4, 5, 8, 9, 10, } =< 4, 5 > S = Q(< 4, 5 >) S = {3, 6, 7, } =< 3, 7 > S = Q(< 3, 7, 11 >) S = {3, 6, 7, 9, 10, } =< 3, 7 > S = Q(< 3, 7 >) S = {2, 4, } =< 2 > S = Q(< 2, 5 >) S = {2, 4, 6, } =< 2 > S = Q(< 2, 7 >) S = {2, 4, 6,..., 2n, } =< 2 > S = Q(< 2, 2n + 1 >) (Universidad de Granada) 7 / 16

8 Problem How to characterize the (finite) subsets A of N + that are a mgs of some qns? Moreover, how many qns s have A as a mgs? For instance, {1} and {2, 3} are not a mgs of a qns. Clearly, if A is a mgs of a qns then 1 A. Furthermore, A must be irreducible, i.e. no element of A can be expressed as a non-trivial sum of elements of A. Let A be a finite irreducible set of N + and denote by Q A the family of qns s that admit A as mgs. Then, clearly, gcd(a) > 1 Q A = ℵ 0. On the other hand, if gcd(a) = 1 then, being N the numerical semigroup generated by A, clearly Q A F (N) > max(a) (Universidad de Granada) 8 / 16

9 Notable elements Let S = {a 1 < a 2 < < a n < } be a non-trivial qns. Numerical definitions Problem A gap of S is an element of G(S) = {1,..., a n + 1} \ {a 1,..., a n }. The Frobenius number of S is the number F (S) = a n + 1 (the largest gap). The gender of S is the number of gaps of S: g(s) = G(S). The embedding dimension of S, denoted by e(s), is the cardinality of the minimal generating set of S. The multiplicity of S is the element m(s) = a 1 (the smallest element of S and of the minimal generating set of S). From a minimal set of generators of a qns S and S, how to/can we compute efficiently F (S)? (Universidad de Granada) 9 / 16

10 Abstract definitions Let S be a qns. Problem The isofrobenius number of S is if (S) = min{f (T ) T is a qns isomorphic to S}. The isogender of S is ig(s) = min{g(t ) T is a qns isomorphic to S}. The isomultiplicity of S is im(s) = min{m(t ) T is a qns isomorphic to S}. How to/can we compute efficiently these numbers? Problem Can these three numbers be obtained from the same qns? Is it unique? (Universidad de Granada) 10 / 16

11 Example Let S = {5, } = Q(< 5, 7, 8, 9, 11 >). Then F (S) = 6, g(s) = 5 and m(s) = 5. On the other hand, clearly, S is isomorphic to T = {2, } = Q(< 2, 5 >) and if (S) = F (T ) = 3, ig(s) = g(t ) = 2 and im(s) = m(t ) = 2. Regarding the embedding dimensions, it is obvious that: Let ϕ : S T be an isomorphism of qns s and let A be the mgs of S. Then ϕ(a) is the mgs of T. In particular, e(s) = e(t ). Problem How to test efficiently if two qns s are isomorphic? Problem Characterize the automorphism group of a qns (Universidad de Granada) 11 / 16

12 We have: Let S and T be two qns s. Let A and B be the mgs s of S and T, respectively. Let A R be a presentation of S. Then, the isomorphisms from S into T are the homomorphisms that extend the bijections f : A B that preserve A R (i.e. such that (u, v) R = f (u) = f (v), for all u, v FS(A), where f : FS(A) T is the canonical homomorphism from the free semigroup FS(A) into T that extends f ). In particular: Let S be a qns, A the mgs of S and A R a presentation of S. Then, the automorphisms of S are the endomorphisms of S that extend the permutations of A that preserve A R (Universidad de Granada) 12 / 16

13 Problem Find a nice presentation for a qns. It is easy to obtain a presentation for a qns S = Q(N) by adding F (N) + 1 relations to a minimal presentation of the numerical semigroup N. Conjecture We obtain a presentation for a qns S = Q(N) by adding m(s) relations to a minimal presentation of the numerical semigroup N. If this is conjecture holds, then we have a presentation for S (that may be considered on its mgs) with less than or equal to m(s)(m(s) + 1)/2 relations. Problem What about minimal presentations for a qns? (Universidad de Granada) 13 / 16

14 Let S be a finite semigroup. We say that S is nilpotent if S n = 1, for some n N +. T.F.A.E. for a finite semigroup S (with zero): S is nilpotent; S satisfies an equation of the form x 1 x n = 0, for some n N + ; S satisfies an equation of the form x n = 0, for some n N + ; S satisfies the pseudoequation x ω = 0. Example Any qns is a commutative nilpotent semigroup. A pseudovariety of semigroups is a class of finite semigroups closed under formation of finite direct products, subsemigroups and homomorphic images. Examples The classes N of nilpotent semigroups, Com of commutative (finite) semigroups and N Com are pseudovarieties of semigroups (Universidad de Granada) 14 / 16

15 Let Num be the class of all qns s (up to isomorphism). Then Num N Com and this inclusion is strict: the semigroup a, b a 2 = b 2 = 0, ab = ba = {a, b, a + b, } is commutative and nilpotent but it is not (isomorphic to) a qns. The class Num is, clearly, closed under formation of subsemigroups but it is not closed under formation of homomorphic images or finite direct products: Let N =< 4, 5 >. Then N = {4, 5, 8, 9, 10, 12 } and so S = Q(N) = {4, 5, 8, 9, 10, }. Clearly, I = {8, 10, } is an ideal of S and S/I is defined by the presentation a, b a 2 = b 2 = 0, ab = ba. Thus, S/I Num. Let S = Q( 2, 5 ) = {2, } and T = Q( 2, 7 ) = {2, 4, }. Then the direct product S T = {[2, 2], [4, 2], [2, 4], [4, 4], [2, 6], } is not (isomorphic) to a qns, since [2, 2] + [2, 2] = [2, 2] + [4, 2] = [4, 2] + [4, 2] (which can not happen in a qns) (Universidad de Granada) 15 / 16

16 Theorem The class Num generates the pseudovariety N Com. Sketch of the proof. Let V be the pseudovariety generated by Num. We suppose that V is strictly contained in N Com. Hence, V must satisfy a non-trivial pseudoequation of the form x α1 1 xn αn = x β1 1 x n βn, with α i, β i N 0 {ω} and x 1,..., x n not necessarily distinct. The proof follows by finding a qns which does not satisfy this pseudoequation (Universidad de Granada) 16 / 16