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Patterns of ideals of numerical semigroups 5 1 Klara Stokes 0 2 School of Engineering Sciences n University of Skövde, Box 408 a 54128 Skövde Sweden J [email protected] 8 2 ] Abstract A R This article introduces patterns of ideals of numerical semigroups, therebyunifyingpreviousdefinitionsofpatternsofnumericalsemigroups. . h Several results of general interest are proved. More precisely, this article at presents results on the structure of the image of patterns of ideals, and m also on thestructure of thesets of patterns admitted by a given ideal. [ 1 Introduction 1 v 1 A numerical semigroup S is a subset of the non-negative integers (denoted by 0 Z ) that contains zero, is closed under addition and has finite complement in 3 + Z . The set of non-zero elements in S is denoted by M(S). 7 + 0 Elements inthe complementZ+\S arecalledgaps, andthe number ofgaps . is the genus of S. The smallest element of M(S) is the multiplicity of S and 1 it is denoted by m(S). The largest integer not in S is the Frobenius element 0 5 andis denotedby F(S). The number F(S)+1is calledthe conductor ofS and 1 is denoted by c(S). An integer x 6∈ S is pseudo-Frobenius if x+s ∈ S for all : s ∈ M(S). The set of pseudo-Frobenius integers is denoted by PF(S). Note v i that F(S)∈PF(S) and that F(S) is the maximum of the elements in PF(S). X It can be provedthat, given a numerical semigroupS, there exists a unique r minimalsetofelementsB ⊂M(S)suchthatanyelementinS canbeexpressed a as a linear combination of elements from B. The elements in B are called minimal generators of S and they are exactly the elements of M(S) that can not be obtained as the sum of two elements of M(S). The cardinality of B is always finite. More precisely it is always less or equal to the multiplicity of S. A numerical semigroup has maximal embedding dimension if the number of minimal generators equals the multiplicity. Arelativeideal ofanumericalsemigroupS isasetH ⊆ZsatisfyingH+S ⊆ H and H+d⊆S for some d∈S. A relative ideal contained in S is an ideal of S. An ideal is proper if it is distinct from S. The set of proper ideals of S has a maximal element with respect to inclusion. This ideal is called the maximal 1 ideal of S, and equals M(S), the set of non-zero elements of S. The dual of a relative ideal H is the relative ideal H∗ =(S−H)={z ∈Z:z+H ⊆S}. Apattern admittedbyanidealI ofanumericalsemigroupS isamultivariate polynomial function p(X ,...,X ) which returns an element p(s ,...,s ) ∈ S 1 n 1 n when evaluated on any non-increasing sequence (s ,...,s ) of elements in I. 1 n We say that the ideal I admits the pattern. If I = S, then we say that the numerical semigroup S admits the pattern. Note that a pattern admitted by an ideal I of a numerical semigroup S is also admitted by any ideal J ⊆I. We will identify the pattern with its polynomial, and, for example, say that the pattern is linear and homogeneous, when the pattern polynomial is linear and homogeneous. The length of a pattern is the number of indeterminates and its degree is the degree of the pattern polynomial. One pattern p induces anotherpatternqifanyidealofanumericalsemigroupthatadmitspalsoadmits q. Two patterns areequivalentif they induce eachother. If any idealsatisfying a given condition c that admits p also admits q, then we say that p induces q undertheconditionc. Twopatternsareequivalentundertheconditioncifthey induce each other under the condition c. Homogeneous linear patterns admitted by numerical semigroups were introduced by Bras-Amoro´s and Garc´ıa-Sánchez in [4]. The patterns that they consideredwerealldefinedbyhomogeneouslinearmultivariatepolynomialswith the whole numerical semigroup as domain. Examples of homogeneous patterns are the homogeneous linear patterns with positive coefficients. It is easy to see that these patterns are admitted by any numerical semigroup. Arf numerical semigrouparecharacterizedbyadmittingthehomogeneouslinear“Arfpattern” X +X −X . Homogeneouslinear patterns of the formX +···+X −X 1 2 3 1 k k+1 generalise the Arf pattern and are called subtraction patterns [4]. The definition of pattern from S to S does not allow for non-homogeneous patterns with constant term outside S. To overcome this problem, when the non-homogeneouspatternswereintroducedin[5],itwaswithM(S)asdomain. Note that with this definition X +a with a ∈ PF(S) is a non-homogeneous linear pattern admitted by S. Admitting the non-homogeneous linear pattern X +X −m(S)isequivalenttothepropertyofmaximalembeddingdimension. 1 2 Since this pattern is induced by the pattern X +X −X , this implies that a 1 2 3 numerical semigroup that is Arf is always of maximal embedding dimension. Further examples of non-homogeneous linear patterns can be found in the numericalsemigroupsassociatedtotheexistenceofcombinatorialconfigurations (see [6]). It was proved in [13, 14] that such numerical semigroups admit the patterns X +X −n for n∈{1,...,gcd(r,k)} and X +···+X +1, 1 2 1 rk/gcd(r,k) where r and k are positive integers that depend on the parameters of the combinatorial configuration. This example motivates the study of a set of patterns that are admitted simultaneously by the same numerical semigroup. Anotherexampleofanon-homogeneouslinearpatternisqX −qm(S),which 1 is admitted by a Weierstrass semigroup S of multiplicity m(S) of a rational place of a function field over a finite field of cardinality q, for which the Geil- Matsumoto bound and the Lewittes’ bound coincide [3]. Similarly, the pattern (q−1)X −(q−1)m(S) is admitted if and only if the Beelen-Ruano’s bound 1 2 equals 1+(q−1)m [5]. Patterns can be used to explore the properties of the numerical semigroup admitting them. For example, the calculations of the formulae for the no- table elements of Mersenne numerical semigroups in [10] rely on the fact that all Mersenne numerical semigroups generated by a consecutive sequence of Mersenne numbers admit the non-homogeneous pattern 2X +1. Similarly, 1 thenon-homogeneouspatternsadmittedbynumericalsemigroupsassociatedto the existence of combinatorial configurations were used to improve the bounds on the conductor of these numerical semigroups. In this article we study patterns of ideals of numerical semigroups. Section2containsbasicresultsaboutthepropertiesoftheimageofpatterns. For example, it is provedthat if the greatestcommon divisor of the coefficients of the pattern p is one and I is an ideal of a numerical semigroup S, then the image p(I) of a pattern is always an ideal of a numerical semigroup. Also, sufficient conditions are given for when p(I)⊆S. Section3presentsanupperboundofthesmallestelementcinp(I)suchthat all integers larger than c belong to p(I), under the condition that the greatest common divisor of the coefficients of p is one. By dividing p by the greatest commondivisorofits coefficients,this resultmakesit possible to calculate p(I) for any admissible pattern p. Section 4 introduces the concepts endopattern and surjective pattern of an ideal, and gives some sufficient and necessary conditions on patterns to have these properties. In Section 5, we generalize the notion of closure of a numerical semigroup with respect to a homogeneous linear pattern to the closure of an ideal of a numerical semigroup with respect to a non-homogeneous linear pattern. We also prove a necessary condition for when the closure of an ideal with respect to a non-homogeneous pattern can be calculated by repeatedly applying the pattern. In Section 6 we prove that the set of patterns admitted by an ideal of a numerical semigroup has the structure of a semigroup, semiring or semiring algebra, depending on if the length and the degree of the patterns is fixed. Section 7 introduces a generalizationofpseudo-Frobenius asa useful toolin the analysis of the structures defined in Section 6. Section 8 introduces infinite chains of ideals of numerical semigroups where the subsequent ideal I is the image of the preceding ideal I under a pattern i i−1 p which is admitted by the first pattern in the chain, and hence by them all. Finally, Section 9 defines polynomial composition of patterns, hence pro- viding yet another operation that creates a pattern admitted by an ideal from several patterns admitted by that ideal. 2 The image of a pattern ApatternpadmittedbyanidealI ofanumericalsemigroupS returnselements inS whenevaluatedoverthenon-increasingsequencesofelementsofI. Wewill 3 now study the image p(I) of I under p. We will need the following well-known result. Lemma 1. Let A⊆Z be closed under addition. Then A does not have finite + complement in Z if and only if A⊆uZ for some u>1. + Proof. If A does not have finite complement, then A does not contain x,y such that gcd(x,y) = 1, since otherwise the maximal ideal of the numerical semigroup hx,yi, which has finite complement, would be contained in A. Therefore gcd(A)=u for some u>1 so that A⊆uZ. If uZ is an ideal of Z with u > 1 and A ⊆ uZ, then clearly |Z \ A| is + infinite. Lemma2. IfI is an ideal ofsomenumericalsemigroup S,thenthereis ac∈I such that z ∈I for all z ∈Z with z ≥c. Proof. IfI is anidealofsomenumericalsemigroupS,then I+S ⊆I,implying that |Z \I|<∞. Therefore there is a c∈I such that z ∈I for all z ∈Z with + z ≥c. If, in Lemma 2, I = S, then the integer c is the conductor of S. If I is a proper ideal, then we call c the maximum of the small elements of I. Theorem 3. Let p(X ,...,X )=a X +···+a X be a homogeneous linear 1 n 1 1 n n 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 ,...,a )=1. 1 n Proof. Assume that gcd(a ,...,a )=1 andlet c be the maximumofthe small 1 n elements of I (see Lemma 2). Let u > 1 and s ∈ I ∩uZ with s ≥ c. Then s+1,s+u ∈ I, with s+1 6∈ uZ, s+u ∈ uZ and s+u > s+1 > s. Since gcd(a ,...,a ) = 1 there is an i ∈ [1,n] such that a is not a multiple of u. 1 n i Therefore i−1a (s + u) + a (s + 1) + n a s ∈ p(I) \ uZ. Lemma 1 j=1 j i j=i+1 j implies that p(I) has finite complement in Z . + P P Notethatifx ,...,x andy ,...,y arenon-increasingsequencesofI,then 1 n 1 n so is x +y ,...,x +y . Since the pattern p is linear and homogeneous we 1 1 n n have p(x ,...,x )+p(y ,...,y ) = p(x +y ,...,x +y )∈ p(I) for all non- 1 n 1 n 1 1 n n increasing sequences x ,...,x ∈I and y ,...,y ∈I, that is, a+b∈p(I) for 1 n 1 n all a,b∈p(I). Hence p(I) is closed under addition. (That linearity of p implies that p(I) is closed under addition was first noted in [4].) Together, the above imply that if 0 ∈ p(I), then p(I) is a numerical semigroup,andif06∈p(I),thenp(I)isthemaximalidealofthenumericalsemigroup p(I)∪{0}. In any case, p(I) is an ideal of a numerical semigroup. Now assume that gcd(a ,...,a )=u>1. Then clearly p(I)⊆uZ, so that 1 n p(I) does not have finite complement in Z and can not be the ideal of any + numerical semigroup. Note that in Theorem3, either p(I) is a numericalsemigroup,or p(I) is the maximalidealofthenumericalsemigroupp(I)∪{0},dependingonwhether0∈ p(I) or not. When I is a proper ideal, then 0∈p(I) exactly when n a =0 i=1 i (seeProposition18). WhenI =S,thenTheorem3impliesthefollowingresult. P 4 Corollary 4. Let p(X ,...,X )=a X +···+a X be a homogeneous linear 1 n 1 1 n n pattern admitted by Z and let S be a numerical semigroup. Then p(S) is a + numerical semigroup if and only if gcd(a ,...,a )=1. 1 n Proof. Apply Theorem 3 with I =S and note that p(0,...,0)=0∈p(S). Clearly the numerical semigroupp(S) is contained in the originalnumerical semigroup S if and only if p is admitted by S. Following [4], a linear homogeneous pattern p(X ,...,X ) = n a X is 1 n i=1 i i premonic if n′ a =1forsomen′ ≤n. Wesaythatalinearnon-homogeneous i=1 i P pattern p(X ,...,X ) = n a X +a is premonic if p(X ,...,X )−a is P1 n i=1 i i 0 1 n 0 premonic. If a =1 then p is monic and so all monic patterns are premonic. 1 P Lemma 5. If p is a premonic linear homogeneous pattern admitted by a numerical semigroup S, then p(S)=S. Proof. If p is a linear homogeneous pattern admitted by S and p is premonic, then n′ a s+ n a 0 = s for all s ∈ S, so that S ⊆ p(S). Clearly i=1 i j=n′+1 j p(S)⊆S, so that p(S)=S. P P Moreover, the image of a premonic linear pattern, homogeneous or not, admitted by an ideal I of a numerical semigroup S, is an ideal of S. Lemma 6. Let p be a premonic linear pattern admitted by an ideal I of a numerical semigroup S. Then p(I) is an ideal of S. Proof. Clearly,ifpisapatternadmittedbyI thenp(I)⊆S. Ifp(X ,...,X )= 1 n n a X +a is a premonic pattern then n′ a = 1 for some 1 ≤ n′ ≤ n. i=1 i i 0 i=1 i If s ,...,s is a non-increasing sequence of elements from I and s ∈ S then 1 n P P s1+s,...,sn′ +s,sn′+1,...,sn is a non-increasing sequence of elements from I for any 1 ≤ n′ ≤ n. We have p(s ,...,s ) + s = n a s + a + s = 1 n i=1 i i 0 n a s +a +( n′ a )s = n′ a (s +s)+ n a s +a = p(s + i=1 i 1 0 i=1 i i=1 i i i=n′P+1 i i 0 1 s,...,sn′+s,sn′+1,...,sn)∈p(I)for all non-increasingsequences s1,...,sn of P P P P elements from I and for all s ∈ S. Therefore p(I)+S ⊆ p(I), and p(I) is an ideal of S. Note thatifp(X ,...,X )= n a X isa premoniclinearpatternadmit- 1 n i=1 i i tedbythemaximalidealM(S)ofanumericalsemigroupsS,thengcd(a ,...,a )= 1 n P 1,sothat,byTheorem3, p(M(S))iseitheranumericalsemigroupcontainedin S or the maximal ideal of a numericalsemigroupcontained in S. But although an ideal of S that contains zero must be equal to S, it is not true in general that if p is a premonic, homogeneous linear pattern admitted by M(S), then p(M(S))∪{0} = S. Consider for example the pattern X +X with image 1 2 2M(S) ( M(S). However, as we have already seen, if p is a premonic linear homogeneous pattern admitted by a numerical semigroup S, then p(S)=S. Example 7. The Arf pattern p(X ,X ,X )=X +X −X is a monic linear 1 2 3 1 2 3 homogeneous pattern. If S is a numerical semigroup Arf, then p(S) = S, and 5 since p(0,0,0)=0 and p−1(0)=(0,0,0)also p(M(S))=M(S). If I is an ideal of S, then I ⊆p(I), but in general it is not true that p(I)⊆I. For example, if S =h3,5,7i and I =S\{0,7}, then p(5,5,3)=7∈p(I)\I and p(I)=M(S). Finally,weshowthatthereisarelationbetweenrelativeidealsandpremonic linear non-homogeneous patterns. Lemma 8. Let p(X ,...,X ) = a X + ··· + a X + a be a linear non- 1 n 1 1 n n 0 homogeneous pattern admitted by an ideal I of a numerical semigroup S and let q(X ,...,X ) = p(X ,...,X )−a be the homogeneous linear part of p. If p 1 n 1 n 0 is premonic (and therefore also q), then q(I) is a relative ideal of S. Proof. Since p(X ,...,X ) = q(X ,...,X )+a by Lemma 6 we have q(I)+ 1 n 1 n 0 S ⊆q(I) and q(I)+a ⊆S. 0 3 Calculating the image of a pattern The following result is useful for calculating the image p(I) of an admissible linear homogeneous pattern. Lemma 9. Let I be an ideal of a numerical semigroup. Let p(X ,...,X ) = 1 n n a X be a homogeneous linear pattern with gcd(a ,...,a ) = d(≥ 1) and i=1 i i 1 n let b ,...,b be (non-unique) integers such that a b +···+a b = d. It is 1 n 1 1 n n P not assumed that p is admitted by I. Then p(s ,...,s ) + d ∈ p(I) for all 1 n non-increasing sequences s ,...,s ∈ I such that s +b ,...,s +b is also a 1 n 1 1 n n non-increasing sequence of elements from I. Proof. Wehavep(s ,...,s )+d=p(s ,...,s )+p(b ,...,b )=p(s +b ,...,s + 1 n 1 n 1 n 1 1 n b ). Therefore, if s +b ,...,s +b is a non-increasing sequence of elements n 1 1 n n from I, then p(s ,...,s )+d∈p(I). 1 n Note that any choice of b ,...,b such that a b +···+a b = d will do. 1 n 1 1 n n In practice it may be useful to instead require that s ≥ ··· ≥ s ≥ c(I) and 1 n s +b ≥···≥s +b ≥c(I) where c(I) is the smallestelement in I such that 1 1 n n z ∈I forallintegersz ≥c. Clearlythenboths ,...,s ands +b ,...,s +b 1 n 1 1 n n are non-increasing sequences of elements from I. Theorem 10. Let I, c(I), p, d and b ,...,b be as in Lemma 9. Then J = 1 n p(I)/d is an ideal of a numerical semigroup. Let c(J) be the maximum of the small elements of J. Also let α = n a /d. Then c(J) < p(s ,...,s )/d i=1 i 1 n whenever s ≥ c(I)−min(0,(α−1)b ) and s ≥ s +max(0,(α−1)(b −b )) n n i j j i P for 1≤i<n. Proof. If d = gcd(a ,...,a ) then d divides all elements of p(I). Dividing 1 n p(I) by d gives the image of I under the pattern q = p/d = n aiX which i=1 d i has relatively prime coefficients c = ai such that c b +···+c b = 1 and n c =α. Therefore,by Theoreim3,dJ =q(I) is an1id1ealofPsomnesnemigroup. i=1 i Anynon-increasingsequences ,...,s ∈Zwiths ≥c(I)isanon-increasing 1 n n P sequence of elements of I. Note that the b :s can be negative integers. Take i 6 s ≥c(I)−min(0,(α−1)b )ands ≥s +max(0,(α−1)(b −b ))for1≤i<n. n n i j j i Undertheseconditionswehavethats +tb ≥s +tb ≥c(I)forall1≤i≤j ≤n i i j j and 0 ≤ t ≤ α−1, so that q(s +tb ,...,s +tb ) ∈ q(I). Now note that 1 1 n n q(s +x+(t+1)b ,...,s +x+(t+1)b )=q(s +x+tb ,...,s +x+tb )+1for 1 1 n n 1 1 n n all0≤t≤α−1andforallx≥0. Also,q(s +x,...,s +x)=q(s ,...,s )+αx. 1 n 1 n Therefore q(I) contains all integers larger or equal to q(s ,...,s ) with s ≥ 1 n n c(I)−min(0,(α−1)b )ands ≥s +max(0,(α−1)(b −b ))for1≤i<n. n i j j i Theorem 10 implies that the set of non-increasing sequences of I which is neededforcalculatingexplicitly p(I)isfinite. However,inpracticethis number will depend on the choice of b ,...,b . 1 n Alinearpatternp(X ,...,X )= n a X +a iscalledstronglyadmissible 1 n i=1 i i 0 ifthepartialsums n′ a ≥1forall1≤n′ ≤n. (Stronglyadmissiblepatterns i=1 i P were introduced differently in [4], but the two definitions are equivalent). P Lemma 11. Let C be a positive integer constant, and let p(X ,...,X ) = 1 n n a X +a beastronglyadmissible linear pattern. LetY ={(t,s ,...,s ): i=1 i i 0 t 2 n ∀ 2 ≤i≤n s ∈I,∀ 2 ≤i≤n−1 s ≥s } and let Y(C) = Y for P i i i+1 t≤x, t∈I t the smallest x∈I such that p(x,s ,...,s )≥C for all s ,...,s ∈I such that 2 n 2 n S x ≥ s ≥ ···≥ s . Then Y(C) is a well-defined finite set and contains the set 1 n of non-increasing sequences s ,...,s ∈I such that p(s ,...,s )<C. 1 n 1 n Proof. First we prove that if n′ a ≥ 1 for all 1 ≤ n′ ≤ n, then there is i=1 i an x ∈ I such that p(x,s ,...,s ) ≥ C for all s ,...,s ∈ I such that x ≥ 2 n 2 n s ≥ ··· ≥ s . Let s ≥ ··· ≥Ps ≥ 0. Then p(s ,...,s ) = n a s +a = 2 n 1 n 1 n i=1 i i 0 n−1( i a )(s −s )+ n a s +a ≥ n−11·(s −s )+1·s +a = i=1 j=1 j i i+1 j=1 j n 0 i=1 i iP+1 n 0 s +a . Therefore, by taking x=s ≥C−a , one has p(x,s ,...,s )≥C for 1 0 1 0 2 n P P P P all s ,...,s ∈I such that x≥s ≥···≥s , so Y is a well-defined finite set. 2 n 2 n Now note that we have also proved that for all s ,...,s ∈ I with s > x, 1 n 1 p(s ,...,s )≥C. Consequently,ifs=(s ,...,s )isanon-increasingsequence 1 n 1 n of elements of I such that p(s ,...,s )<C, then s∈Y. 1 n The following algorithm calculates p(I) by calculating first an upper bound C ≥c(J) and then calculating p(s) for all s∈Y(C). Lemma 12. Letnotation beas in Lemma9 andTheorem 10. Assumealso that the admissible homogeneous pattern polynomial p(X ,...,X ) = n a X is 1 n i=1 i i strongly admissible. The following algorithm can be used to calculate p(I). . P 1. Set q =p/d. 2. Calculate C = q(s ,...,s ) with s = c(I)+min(0,αb ) and s = s + 1 n n n i j min(0,α(b −b )) for 1≤i<n. j i 3. Calculate Q := {q(s ,...,s ) : (s ,...,s ) ∈ Y(C)} where Y(C) is the 1 n 1 n set defined in Lemma 11. 4. Now q(I)=Q∪{z ∈Z:z ≥C} and p(I)={ds:s∈q(I)}. 7 Proof. By combining Theorem 10 and Lemma 11. Given a linear strongly admissible pattern polynomial and two ideals I of a numerical semigroup S and J of a numerical semigroup S′, step 3 in the algorithminLemma12alonecanbeusedtodeterminewhetherornotp(I)⊆J, after defining C = min(s ∈J : n ∈ J ∀ n ∈ {z ∈ Z: z ≥ s}). In the particular case when I is an ideal of a numerical semigroupS and J =S (and so C is the conductor of S), this calculation determines whether or not I admits p. The existence of an algorithm that determines if a strongly admissible pattern is admitted by a numerical semigroup was first announced in [4]. Lemma 13. Let I be an ideal of a numerical semigroup, let p(X ,...,X ) = 1 n n a X +a be an admissible linear pattern and let i=1 i i 0 P q(X ,...,X )=p(X ,...,X )−a . 1 n 1 n 0 Then p(I)=q(I)+a . 0 Proof. Indeed, any element in p(I) is of the form p(s ,...,s ) = n a s + 1 n i=1 i i a =q(s ,...,s )+a . 0 1 n 0 P Together Lemma 12 and Lemma 13 can be used to calculate the image of an ideal of a numerical semigroup under a linear strongly admissible pattern p in a finite number of steps. 4 Patterns of ideals of numerical semigroups In this article a pattern admitted by an ideal I of a numericalsemigroupS is a multivariate polynomial function which evaluated on non-increasing sequences ofelements fromI returnsanelementofS. This definitiongeneralisesprevious definitions of patterns admitted by numerical semigroups. Indeed, a homoge- neouslinearpatternasdefinedin[4]isaccordingtoourdefinitionstillapattern admitted by a numerical semigroup. However, a non-homogeneous linear pattern as defined in [5] is now a pattern admitted by the maximal ideal of some numerical semigroup. The concept can be generalisedfurther, for example by relaxing the criteria that the codomain of a pattern admitted by an ideal necessarily should be a numerical semigroup containing the ideal. Then the codomain of the pattern canbeanothernumericalsemigroup,orgeneralisingevenmore,anidealofsome numerical semigroup. It is possible to go even further by considering relative ideals instead of ideals. One can also restrict to patterns with some particular property like for example linearity or homogeneity. We say that a linear pattern that returns an element in I when evaluated on the non-increasing sequences of elements of I is an endopattern of I. A patternadmittedbyanidealI withcodomainJ issurjective whenp(I)=J. A surjectiveendopatternofI isthereforeapatternpsuchthatp(I)=I. Formally, 8 an endopattern of I is an endomorphism of the set of non-increasing sequences of n elements from I. Note that, for example, the map x 7→ (x,...,x) is an embedding ofthe image of the patternin the setof non-increasingsequences of n elements from I. In this article, the focus is on linear endopatterns of numerical semigroups and ideals of numerical semigroups, in particular maximal ideal. To avoid con- fusion we will each time explicitly state the properties of the patterns that we consider in each moment. We start with necessary conditions for linear patterns to be endopatterns and surjective endopatterns of numerical semigroups. We will repeatedly make use of the following result, first proved in [4] for homogeneous linear patterns and in [5] for non-homogeneous linear patterns. Here we prove the result for idealsofnumericalsemigroups,makinguseofAbel’spartialsummationformula (this proof is due to Christian Gottlieb). Lemma 14. If p(X ,...,X ) = n a X +a is a linear pattern admitted 1 n i=1 i i 0 by an ideal I of a numerical semigroup S (i.e. p(s ,...,s ) ∈ S for all non- 1 n P increasing sequences s ,...,s of elements from I), then 1 n • n′ a ≥0 for all 1≤n′ ≤n, and i=1 i • Pn a s ≥ n a s for all non-increasing sequences s ,...,s ∈I. i=1 i i i=1 i n 1 n Proof.PAssume thePre is an n′ with 1 ≤ n′ ≤ n such that n′ a < 0. If s is i=1 i very large compared to t we obtain P n′ n p(s,...,s,t,...,t)= a s+ a t<0, i i n′ n−n′ Xi=1  i=Xn′+1 !   implying that p|ca{nzno}t|be{za p}attern admitted by I. Now, let A = j a . By Abel’s formula for summation by parts [1], we j i=1 i have n a (s −s )=A (s −s )− n−1A (s −s )=− n−1A (s − i=1 i i Pn n n n i=1 i i+1 i i=1 i i+1 s )≥0. i P P P Proposition 15. A linear endopattern of a numerical semigroup S is simply a linearpatterndefinedbyapolynomial p(X ,...,X )= n a X +a admitted 1 n i=1 i n 0 by S. Therefore P • n′ a ≥0 for all 1≤n′ ≤n, and i=1 i • Pa ∈S. 0 Proof. LetpbealinearendopatternofS,thenpisapatterndefinedbyalinear multivariate polynomial p(X ,...,X ) = n a X +a such that evaluated 1 n i=1 i i 0 onanynon-increasingsequenceofelementsofS,theresultisinS. Inparticular, P p(0,...,0)=a ∈S. For the rest of the statement, apply Lemma 14. 0 9 Proposition 16. A linear surjective endopattern p of a numerical semigroup S is necessarily homogeneous. If p is a premonic homogeneous endopattern of S, then p is always surjective. Proof. A surjective endopattern p of S is an endopattern of S, therefore, by Lemma15,pisalinearpatterndefinedbyapolynomialoftheformp(X ,...,X )= 1 n n a X +a with a ∈S and n′ a ≥ 0 for all 1≤ n′ ≤ n. But if a >0 i=1 i i 0 0 i=1 i 0 then this gives p(s ,...,s ) > 0 for all non-increasing sequences of S so that 1 n P P p(S)(S. Consequently,ifp(S)=S,thenpisdefinedbyahomogeneouslinear pattern. Finally, if p is premonic then, by Lemma 5, p is surjective. The next result gives a necessary condition for when a polynomial defines a linear pattern admitted by a proper ideal of a numerical semigroup. When theidealisamaximalidealthenthis resultstrengthensthe necessarycondition given in [5]. Lemma 17. If S is a numerical semigroup and p= n a X +a is a linear i=1 i i 0 pattern admitted by a proper ideal I of S, then P • n′ a ≥0 for all 1≤n′ ≤n and, moreover, i=1 i • Pn a ≥max(0,−a /µ(I)), where µ(I)=min(I). i=1 i 0 Proof.PFor the firstpart, apply Lemma 14. The secondpart is animprovement of n a ≥0whichisrelevantonlywhena <0. Therearenolinearpatterns i=1 i 0 admittedbySwitha <0(seeProposition15),buttheremaybelinearpatterns 0 P admitted by I with that property. Therefore assume that p(X ,...,X ) = 1 n n a X +a is a linear pattern admitted by I with a < 0 and n a < i=1 i i 0 0 i=1 i max(0,−a /µ(I)) = −a /µ(I), then p(µ(I),...,µ(I))) = n a µ(I)+a < P 0 0 i=1 i P 0 (−a /µ(I))·µ(I)+a =0 so that p(µ(I),...,µ(I))6∈S. But then p cannot be 0 0 P a pattern of I and we have a contradiction. We nowuseLemma 17to givenecessaryconditionsforwhenapatternisan endopattern of a proper ideal of a numerical semigroup. Proposition 18. A linear endopattern of a proper ideal I of a numerical semigroup S is a linear pattern p(X ,...,X )= n a X +a admitted by I, and 1 n i=1 i i 0 so (by Lemma 17) p necessarily satisfies P • n′ a ≥0 for all 1≤n′ <n, i=1 i • Pn a ≥max(0,−a /µ(I)), i=1 i 0 and adPditionally • a >0 or n a >max(0,−a /µ(I)) (or both), 0 i=1 i 0 where µ(I)=miPn(I). 10