Quotient Numerical Semigroups (work in progress) V´ ıtor Hugo Fernandes FCT-UNL/CAUL (joint work with Manuel Delgado ) February 5, 2010 Iberian meeting on numerical semigroups Granada 2010 2010.02.03-05 (Universidad de Granada) 1 / 16
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. 2010.02.03-05 (Universidad de Granada) 2 / 16
Examples I = N + (“the” trivial qns ) • N = < 1 > = ⇒ Q ( N ) = {∞} , • N = < 3 , 5 > = ⇒ Q ( N ) = { 3 , 5 , 6 , ∞} , I = { 8 , →} 2010.02.03-05 (Universidad de Granada) 3 / 16
• N = < 4 , 5 > = ⇒ Q ( N ) = { 4 , 5 , 8 , 9 , 10 , ∞} , I = { 12 , →} 2010.02.03-05 (Universidad de Granada) 4 / 16
• N = < 4 , 7 > = ⇒ Q ( N ) = { 4 , 7 , 8 , 11 , 12 , 14 , 15 , 16 , ∞} , I = { 18 , →} 2010.02.03-05 (Universidad de Granada) 5 / 16
• 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 , →} 2010.02.03-05 (Universidad de Granada) 6 / 16
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 , . . . , 2 n , ∞} = < 2 > S = Q ( < 2 , 2 n + 1 > ) 2010.02.03-05 (Universidad de Granada) 7 / 16
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 ) . 2010.02.03-05 (Universidad de Granada) 8 / 16
“Notable elements” Let S = { a 1 < a 2 < · · · < a n < ∞} be a non-trivial qns . “Numerical” definitions 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 ). Problem From a minimal set of generators of a qns S and | S | , how to/can we compute “efficiently” F ( S )? 2010.02.03-05 (Universidad de Granada) 9 / 16
“Abstract” definitions Let S be a qns. 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 } . Problem How to/Can we compute “efficiently” these numbers? Problem Can these three numbers be obtained from the same qns ? Is it unique? 2010.02.03-05 (Universidad de Granada) 10 / 16
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 . 2010.02.03-05 (Universidad de Granada) 11 / 16
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 ⇒ ¯ f ( u ) = ¯ that preserve � A | R � (i.e. such that ( u , v ) ∈ R = 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 � . 2010.02.03-05 (Universidad de Granada) 12 / 16
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 ? 2010.02.03-05 (Universidad de Granada) 13 / 16
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. 2010.02.03-05 (Universidad de Granada) 14 / 16
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 ). 2010.02.03-05 (Universidad de Granada) 15 / 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 α 1 1 · · · x α n = x β 1 1 · · · x β n n , n 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. 2010.02.03-05 (Universidad de Granada) 16 / 16
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