Patterns of ideals of numerical semigroups Klara Stokes AMS - PowerPoint PPT Presentation

Patterns of ideals of numerical semigroups Klara Stokes AMS Sectional Meeting Special Session on Factorization and Arithmetic Properties of Integral Domains and Monoids March 24, 2019

Table of Contents Introduction 1 Patterns of ideals of numerical semigroups 2 The image of a pattern 3 Closures of ideals with respect to patterns 4 Generalized pseudo-Frobenius numbers 5 Giving structure to the set of patterns admitted by an ideal 6 Klara Stokes March 24, 2019

Table of Contents Introduction 1 Patterns of ideals of numerical semigroups 2 The image of a pattern 3 Closures of ideals with respect to patterns 4 Generalized pseudo-Frobenius numbers 5 Giving structure to the set of patterns admitted by an ideal 6 Klara Stokes March 24, 2019

Numerical semigroups Denote Z + the set of non-negative integers. A numerical semigroup is a subset S ⊂ Z + , such that S is closed under addition, 0 ∈ S and the complement ( Z + ) \ S is finite. Klara Stokes March 24, 2019

Numerical semigroups The multiplicity of a numerical semigroup is its smallest non-zero element. The conductor of a numerical semigroup is the smallest element such that all subsequent natural numbers belong to the numerical semigroup (the Frobenius number +1). The gaps of a numerical semigroup are the elements in the complement of the numerical semigroup. Klara Stokes March 24, 2019

A relative ideal I of a numerical semigroup S is a set I ⊆ Z satisfying I + S ⊆ I I + d ⊆ S for some d ∈ S . An ideal is a relative ideal contained in S (so d = 0). An ideal is proper if it is distinct from S . Examples The maximal ideal of a numerical semigroup S is M ( S ) = S \ { 0 } . It is maximal among the proper ideals of S . Let S = � 3 , 7 � = { 0 , 3 , 6 , 7 , 9 , 10 , 12 , . . . } : ◮ H = S \ { 0 , 6 } is NOT an ideal, ◮ I = S \ { 0 , 7 } = { 3 , 6 , 9 , 10 , 12 , . . . } is an ideal, ◮ I − 7 = {− 4 , − 1 , 2 , 3 , 5 , . . . } is a relative ideal. The dual of a relative ideal H is the relative ideal H ∗ = ( S − H ) := { x ∈ Z : x + H ⊆ S } . Klara Stokes March 24, 2019

Table of Contents Introduction 1 Patterns of ideals of numerical semigroups 2 The image of a pattern 3 Closures of ideals with respect to patterns 4 Generalized pseudo-Frobenius numbers 5 Giving structure to the set of patterns admitted by an ideal 6 Klara Stokes March 24, 2019

Definition. [Bras-Amor´ os and Garc´ ıa-S´ anchez, 2006] A homogeneous pattern admitted by a numerical semigroup S is a homogeneous linear multivariate polynomial p = � n i =1 a i X i such that p ( s 1 , . . . , s n ) ∈ S for all non-increasing sequences s 1 , · · · , s n ∈ S . Examples. Arf numerical semigroups are characterized by admitting the homogeneous linear “Arf pattern” X 1 + X 2 − X 3 . Homogeneous linear patterns of the form X 1 + · · · + X k − X k +1 generalise the Arf property and are called subtraction patterns [Bras-Amor´ os and Garc´ ıa-S´ anchez, 2006]. Klara Stokes March 24, 2019

But with this definition of pattern all non-homogeneous patterns must have constant term in S . n � p (0 , . . . , 0) = a i · 0 + a 0 = a 0 ∈ S . i =1 To overcome this problem, when the non-homogeneous patterns were introduced it was with M ( S ) as domain [Bras-Amor´ os, Garc´ ıa-S´ anchez, and Vico-Oton,2013]. But now we have two different definitions of patterns. Let us generalise and unify! Klara Stokes March 24, 2019

Definition. A pattern admitted by an ideal I of a numerical semigroup S is a multivariate polynomial function which returns an element in S when evaluated on any non-increasing sequence of elements from I . We say that the ideal I admits the pattern. Klara Stokes March 24, 2019

Definition. A pattern admitted by an ideal I of a numerical semigroup S is a multivariate polynomial function which returns an element in S when evaluated on any non-increasing sequence of elements from I . We say that the ideal I admits the pattern. If I = S , then we say that the numerical semigroup S admits the pattern. What happened with the previous definitions of patterns? Homogeneous patterns evaluted on S have become patterns admitted by S . Non-homogeneous patterns evaluated on M ( S ) have become patterns admitted by M ( S ). Note that a pattern admitted by an ideal I of a numerical semigroup S is also admitted by any ideal J ⊆ I . Klara Stokes March 24, 2019

We identify the pattern with its polynomial. We say that the pattern is linear and homogeneous , when the pattern polynomial is linear and homogeneous. The length of a pattern: the number of indeterminates. The degree of a pattern: the degree of the pattern polynomial. One pattern p induces another pattern q if any ideal of a numerical semigroup that admits p also admits q . Two patterns are equivalent if they induce each other. Example. Consider the Arf pattern p Arf ( X 1 , X 2 , X 3 ) = X 1 + X 2 − X 3 . It induces q ( X 1 , X 2 ) = p Arf ( X 1 , X 1 , X 2 ) = 2 X 1 − X 2 . It was proved by Campillo, Farr´ an and Munuera that q and p Arf are equivalent. Klara Stokes March 24, 2019

Table of Contents Introduction 1 Patterns of ideals of numerical semigroups 2 The image of a pattern 3 Closures of ideals with respect to patterns 4 Generalized pseudo-Frobenius numbers 5 Giving structure to the set of patterns admitted by an ideal 6 Klara Stokes March 24, 2019

Theorem Let p ( X 1 , . . . , X n ) = a 1 X 1 + · · · + a n X n be a homogeneous linear pattern admitted by Z + and let I be an ideal of a numerical semigroup S. Then p ( I ) is an ideal of some numerical semigroup if and only if gcd( a 1 , . . . , a n ) = 1 . In particular, if p ( X 1 , . . . , X n ) = a 1 X 1 + · · · + a n X n is a homogeneous linear pattern admitted by Z + and S a numerical semigroup, then p ( S ) is a numerical semigroup if and only if gcd( a 1 , . . . , a n ) = 1. Klara Stokes March 24, 2019

Lemma If I is an ideal of some numerical semigroup S, then there is a c ∈ I such that z ∈ I for all z ∈ Z with z ≥ c. We call this c the maximum of the small elements of I . Theorem Let I be an ideal of a numerical semigroup, p ( X 1 , . . . , X n ) = � n i =1 a i X i a homogeneous linear pattern with gcd( a 1 , . . . , a n ) = d ( ≥ 1) and let b 1 , . . . , b n (non-unique) integers such that a 1 b 1 + · · · + a n b n = d. Then J = p ( I ) / d is an ideal of a numerical semigroup. Klara Stokes March 24, 2019

Lemma If I is an ideal of some numerical semigroup S, then there is a c ∈ I such that z ∈ I for all z ∈ Z with z ≥ c. We call this c the maximum of the small elements of I . Theorem Let I be an ideal of a numerical semigroup, p ( X 1 , . . . , X n ) = � n i =1 a i X i a homogeneous linear pattern with gcd( a 1 , . . . , a n ) = d ( ≥ 1) and let b 1 , . . . , b n (non-unique) integers such that a 1 b 1 + · · · + a n b n = d. Then J = p ( I ) / d is an ideal of a numerical semigroup. Let c ( J ) be the maximum of the small elements of J and let α = � n i =1 a i / d. Then c ( J ) < p ( s 1 , . . . , s n ) / d whenever s n ≥ c ( I ) − min(0 , ( α − 1) b n ) and s i ≥ s j + max(0 , ( α − 1)( b j − b i )) for 1 ≤ i < n. Klara Stokes March 24, 2019

Therefore, the set of non-increasing sequences of I which is needed for calculating explicitly p ( I ) is finite. A linear pattern p ( X 1 , . . . , X n ) = � n i =1 a i X i + a 0 is called strongly i =1 a i ≥ 1 for all 1 ≤ n ′ ≤ n . admissible if the partial sums � n ′ I have written an algorithm for calculating p ( I ) when p is strongly admissible. This algorithm is available in the numerical semigroup package NumericalSgps of GAP. Klara Stokes March 24, 2019

The image of linear patterns are (essentially) ideals. Do all ideals and all numerical semigroups appear in this way? Proposition Any numerical semigroup S = � a 1 , . . . , a e � is the image of Z + under the homogeneous pattern p ( X 1 , . . . , X e ) = a 1 X 1 + � e i =2 ( a i − a i − 1 ) X i . So it is possible to define a numerical semigroup in terms of a pattern! Klara Stokes March 24, 2019

If the numerical semigroup S ′ is the image of a numerical semigroup S ⊇ S ′ under a pattern p admitted by S , then S ′ admits p . Consider the chain of numerical semigroups S ⊇ p ( S ) ⊇ p ( p ( S )) ⊇ · · · . If this chain stabilizes, then it does so immediately and S = p ( S ). Otherwise, what can we say about how S and p ( S ) relate? Klara Stokes March 24, 2019

The quotient of a numerical semigroup S by a positive integer d is the numerical semigroup S d = { x ∈ Z + : dx ∈ S } . Lemma Let S be a numerical semigroup and let p ( X 1 , X 2 ) = a 1 X 1 + a 2 X 2 be a linear homogeneous pattern in two variables (not necessarily admitted by p ( S ) S) such that a 1 ∈ S and gcd( a 1 , a 2 ) = 1 . Then S = a 1 + a 2 . Corollary Any numerical semigroup S is the quotient from division by d of infinitely many numerical semigroups of the form p ( S ) for some pattern p, for any d ∈ Z , d ≥ 2 . Klara Stokes March 24, 2019

Table of Contents Introduction 1 Patterns of ideals of numerical semigroups 2 The image of a pattern 3 Closures of ideals with respect to patterns 4 Generalized pseudo-Frobenius numbers 5 Giving structure to the set of patterns admitted by an ideal 6 Klara Stokes March 24, 2019