MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES. hal , version 1-22 Apr 2014

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1 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES JONATHAN CHAPPELON*, IGNACIO GARCÍA-MARCO, LUIS PEDRO MONTEJANO, AND JORGE LUIS RAMÍREZ ALFONSÍN Abstract. In this paper, we investigatethe Möbius function µ S associated to a (locally finite) poset arising from a semigroup S of Z m. For, we introduce and develop a new approach to study µ S by using the Hilbert series associated to S. The latter allow us to provide formulas for µ S when S is a semigroup with unique Betti element, and when S is a complete intersection numerical semigroup with three generators. We also give a characterization for a finite locally poset to be isomorphic to a semigroup poset. With this in hand, we are able to calculate the Möbius function of certain posets (for instance the classical arithmetic Möbius function) by computing the Möbius function of the corresponding semigroup poset. 1. Introduction The Möbius function is an important concept associated to (locally finite) posets. Möbius function can be considered as a generalization of the classical Möbius arithmetic function onthe integers (given by the Möbius function of the poset obtained fromthe positive integers partially ordered by the divisibility). Möbius function has been extremely useful to investigate many different problems. For instance, the inclusion-exclusion principle can be retrieved by considering the set of all subsets of a finite set partially ordered by inclusion. We refer the reader to [19] for a large number of applications of the Möbius function. In this paper, we investigate the Möbius function associated to posets arising naturally from subsemigroups of Z m as follows. Let a 1,...,a n be nonzero vectors of Z m and let S = a 1,...,a n denote the semigroup generated by a 1,...,a n, that is, S = a 1,...,a n = {x 1 a 1 + +x n a n x 1,...,x n N}. We say that S is pointed if S ( S) = {0}, where S := { x x S}. Whenever S is pointed, S induces on Z m a structure of poset whose partial order S is defined by x S y y x S for all x and y in Z m. This (locally finite) poset will be denoted by (Z m, S ). We denote by µ S the Möbius function associated to (Z m, S ). As far as we are aware, µ S has only been investigated when S is a numerical semigroup, i.e., when S N and gcd{a 1,...,a n } = 1. Moreover, the only known results concerning µ S is an old theorem due to Deddens [5] that determines the value of µ S when S has exactly two generators, and a recent paper due to Chappelon and Ramírez Alfonsín [4] where the authors investigate µ S when S = a,a+d,...,a+kd with a,k,d Z +. In both papers the authors approach the problem by a thorough study of the intrinsic properties of each semigroup. It is worth pointing out that the results and techniques developed in this paper are deeply inspired by those of [4]. Date: April 22, Mathematics Subject Classification. 20M15; 05A99; 06A07; 11A25; 20M05; 20M25. Key words and phrases. Möbius function, locally finite poset, semigroup, Hilbert series, denumerant. * Corresponding Author: Phone/Fax: jonathan.chappelon@um2.fr. 1

2 2 J. CHAPPELON, I. GARCÍA-MARCO, L.P. MONTEJANO, AND J.L. RAMÍREZ ALFONSÍN In this work we introduce and develop a different and more general approach to the study of µ S by means of the Hilbert series of the semigroup S. As a consequence of this point of view we are able to provide formulas for µ S when S belongs to some families of semigroups. Finally, we study when a locally finite poset is isomorphic to a semigroup poset and, thus, its Möbius function can be computed by means of the techniques introduced in this work. This paper is organized as follows. In the next section, we review some classic notions of the Möbius function and apply them to the particular case of semigroup posets. In Section 3, we recall some definitions and results about Hilbert series and prove two general results relating the Möbius function of (Z m, S ) and the Hilbert series of S for every pointed semigroup S. The latter are the key results that will be used in Section 4 to provide explicit formulas for µ S when S belongs to certain families of semigroups; namely, when S = N m, when S is a semigroup with a unique Betti element and when S = a 1,a 2,a 3 N is a complete intersection numerical semigroup. Finally, in Section 5, we provide a criterion for determining when a locally finite poset is isomorphic to a semigroup poset and, thus, its Möbius function can be computed by means of the Möbius function of its corresponding semigroup poset. In particular, we are able to recover the classical arithmetic Möbius function and the Möbius function of the set of all subsets of a finite set partially ordered by inclusion. 2. Möbius function associated to a semigroup poset Let (P, ) be a partially ordered set, or poset for short. The strict partial order < P is the reduction of P given by a < P b if and only if a P b and a b. For any a and b in the poset P, the segment between a and b is defined by [a,b] P := {c P a P c P b}. A poset is said to be locally finite if every segment has finite cardinality. In this paper, we only consider locally finite posets. Let a and b be elements of the poset P. A chain of length l 0 between a and b is a subset of [a,b] P containing a and b, with cardinality l+1 and totally ordered by < P, that is {a 0,a 1,...,a l } [a,b] P such that a = a 0 < P a 1 < P a 2 < P < P a l 1 < P a l = b. For any nonnegative integer l, we denote by C l (a,b) the set of all chains of length l between a and b. The cardinality of C l (a,b) is denoted by c l (a,b). This number always exists because the poset P is supposed to be locally finite. For instance, the number of chains c 2 (2,12), where the poset is the set N partially ordered by divisibility, is equal to 2. Indeed, there are exactly 2 chains of length 2 between 2 and 12 in [2,12] N = {2,4,6,12}, which are {2,4,12} and {2,6,12}. For any locally finite poset P, the Möbius function µ P is the integer-valued function on P P defined by (1) µ P (a,b) = l 0 ( 1) l c l (a,b), for all elements a and b of the poset P. One can remark that this sum is always finite because, for a and b given, there exists a maximal length of a possible chain between a and b since the segment [a,b] P has finite cardinality.

3 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES 3 The concept of Möbius function for a locally finite poset (P, ) was introduced by Rota in [19]. In this paper, Rota proves the following property of the Möbius function: for all (a,b) P P, (2) µ P (a,a) = 1 and µ P (a,c) = 0. c [a,b] P In this work we will consider posets associated to semigroups of Z m. We will begin by summarizing some generalities on semigroups that are useful for the understanding of this work. We refer the readers to [3] or, more generally, to [15] for further details. Let S := a 1,...,a n Z m denote the subsemigroup of Z m generated by a 1,...,a n Z m, i.e., S := a 1,...,a n = {x 1 a 1 + +x n a n x 1,...,x n N}. The semigroup S induces the binary relation S on Z m given by x S y y x S. It turns out that S is an order if and only if S is pointed. Moreover, whenever S is pointed the poset (Z m, S ) is locally finite. We denote by µ S the Möbius function associated to (Z m, S ). It is easy to see that µ S can be considered as a univariate function of Z m. Indeed, for all x,y Z m and for all l 0, we have that (3) c l (x,y) = c l (0,y x). The above follows since the set C l (x,y) is in bijection with C l (0,y x). Indeed the map that assigns the chain {x 0,x 1,...,x l } C l (x,y) to the chain {0,x 1 x 0,...,x l x 0 } C l (0,y x) is clearly a bijection. Thus, by definition of µ S and equality (3) we obtain for all x,y Z. µ S (x,y) = µ S (0,y x) In the sequel of this paper we shall only consider the reduced Möbius function µ S : Z m Z defined by µ S (x) := µ S (0,x), for all x Z m. Thus, the formula given by (2) can be more easily presented when the locally finite poset is (Z m, S ). Proposition 2.1. Let S be a pointed semigroup and let x Z m. Then, { 1 if x = 0, µ S (x b) = 0 otherwise. b S Proof. Firstly, from (1) we observe that µ S (b) = 0 for all b / S. Since S is pointed, if b S then b / S and, hence, b S µ S(0 b) = µ S (0) = 1. Take x 0, then from (2) 0 = b [0,x] Z mµ S (b) = b S x b S µ S (b) = b S x b S µ S (x b) = b S µ S (x b). The formula presented in Proposition 2.1 will be very useful to get most or our results.

4 4 J. CHAPPELON, I. GARCÍA-MARCO, L.P. MONTEJANO, AND J.L. RAMÍREZ ALFONSÍN 3. The Hilbert and Möbius series In this section we present two results (Theorem 3.2 and Theorem 3.3), both relating the Möbius function of the poset (Z m, S ) with the Hilbert series of the semigroup S. Before proving these theorems we recall some basic notions of multivariate Hilbert series. For a thorough study of multivariate Hilbert series we refer the reader to the book by Kreuzer and Robbiano [13]. Let k be any field and let S = a 1,...,a n be a subsemigroup of Z m. The semigroup S inducesagradingintheringofpolynomialsk[x 1,...,x n ]byassigningdeg S (x i ) := a i forall i {1,...,n}. Then the S-degree of themonomial m := x α 1 1 xαn n is deg S(m) := α i a i ; we say that a polynomial is S-homogeneous if all its monomials have the same S-degree. For all b Z m, we denote by k[x 1,...,x n ] b the k-vector space formed by all polynomials S-homogeneous of S-degree b. Consider I k[x] an ideal generated by S-homogeneous polynomials, then for all b Z m we denote by I b the k-vector space formed by the S-homogeneous polynomials of I of S-degree b. Note that I b is a k-vector subspace of k[x 1,...,x n ] b. The quotient ring k[x 1,...,x n ]/I is also S-graded by taking (k[x 1,...,x n ]/I) b := k[x 1,...,x n ] b /I b for all b Z m. Whenever S is pointed the k-vector space k[x 1,...,x n ] b has finite dimension for all b Z m by [13, Proposition ]. Hence, one can define the multigraded Hilbert function of M := k[x 1,...,x n ]/I as HF M : Z m N, where HF M (b) := dim k (M b ) = dim k (k[x 1,...,x n ] b ) dim k (I b ) for al b Z m. For every b = (b 1,...,b m ) Z m, we denote by t b the monomial t b 1 1 t bm m in the Laurent polynomial ring Z[t 1,...,t m,t 1 1,...,t 1 m ]. We define the multivariate Hilbert series of M as the following formal power series in Z[[t 1,...,t m,t 1 1,...,t 1 m ]]: H M (t) := HF M (b)t b. b Z m For every S-homogeneous ideal I, the Hilbert series of M = k[x 1,...,x n ]/I can be expressed as a quotient of polynomials in the Laurent polynomial ring in the following way (see [13, Theorem ]): H M (t) = where α Z m and h(t 1,...,t m ) Z[t 1,...,t m ]. t α h(t 1,...,t m ) (1 t a 1 ) (1 t a n), Denote by I S the toric ideal of S, i.e., the kernel of the k-homomorphism ϕ : k[x 1,...,x n ] k[t 1,...,t m,t 1 1,...,t 1 m ] induced by ϕ(x i ) = t a i for all i {1,...,n}. It is well known that I S is S-homogeneous (see [20, Corollary 4.3]). Therefore, it makes sense to study the multivariate Hilbert series of k[x 1,...,x n ]/I S with respect to the grading induced by S. Proposition 3.1. Let S be a pointed semigroup and M := k[x 1,...,x n ]/I S. Then, H M (t) = b S tb.

5 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES 5 Proof. Take b Z m, then k[x 1,...,x n ] b = {0} and HF M (b) = 0 whenever b / S. Let us prove that HF M (b) = 1 for all b S. Indeed, ϕ induces an isomorphism of k-vector spaces between M b and k[t b ], for all b S. Hence, HF M (b) = dim k (M b ) = dim k (k[t b ]) = 1. Fromnowon, we denotebyh S (t)themultivariatehilbert series ofk[x 1,...,x n ]/I S and we call it the Hilbert series of S. We may now state and prove Theorems 3.2 and 3.3 that relate µ S with the Hilbert series of S. The proofs of both results rely on Proposition 2.1. Theorem 3.2. Let c 1,...,c k be nonzero vectors of Z m and denote (1 t c 1 ) (1 t c k ) H S (t) = b Z m f b t b Z[[t 1,...,t m,t 1 1,...,t 1 m ]]. Then, b Z m f b µ S (x b) = 0 for all x / { i A c i A {1,...,k}}. Proof. Firstly, from Proposition 3.1, we obtain that f b = ( 1) A {1,...,k} b i A c i S for all b Z m. Set := { i A c i A {1,...,k}}. By Proposition 2.1, for all x / and A {1,...,k} we have that ( µ S x ) a i b = 0. b S i A Hence, for all x / it follows that α b µ S (x b) = b Z m b S A {1,...,k} A, ( 1) A µ S ( x i A where α b = A {1,...,k} b i A c i S ( 1) A = f b, which completes the proof. c i b We notice that the formula (1 t c 1 ) (1 t c k ) HS (t) = b Z m f b t b might have an infinitenumber of terms. Nevertheless, forevery x Z m theformula b Z m f b µ S (x b) = 0 only involves a finite number of nonzero summands due to the fact that µ S (x b) 0 implies that x b S and S is pointed. As a consequence of this result, whenever we know an explicit expression of H S (t) as H S (t) = f(t) (1 t c 1 ) (1 t c k), where f(t) Z[t 1,...,t m,t 1 1,...,t 1 m ] for some c 1,...,c k Z m, we can derive a recursive formula for the Möbius function. Although this will be done for some semigroups, unfortunately we cannot do it systematically. For instance, in the case of the so called almost arithmetic semigroups, i.e., subsemigroups S = a 1,...,a n N such that all but one of the a i form an ordinary arithmetic progression. In [17, Theorem 3] the authors find an expression of the Hilbert series of these semigroups as a quotient of two polynomials needing some previously computed parameters. In this case, the recursive formula for the Möbius function might be very difficult to compute. We rather prefer to avoid this long and tedious calculation. ) = 0,

6 6 J. CHAPPELON, I. GARCÍA-MARCO, L.P. MONTEJANO, AND J.L. RAMÍREZ ALFONSÍN Now, we consider the Möbius series G S, the generating function of the Möbius function, which is the power series in Z[[t 1,...,t m,t 1 1,...,t 1 m ]] defined by G S (t) := µ S (b)t b. b Z m Theorem 3.3. Let S be a pointed semigroup. Then, H S (t).g S (t) = 1. Proof. From the definitions of H S (t) and G S (t), we obtain that ( )( ) H S (t).g S (t) = t b µ S (b)t b = ( ) µ S (b c) t b. b S b Z m b Z m c S The result follows by Proposition 2.1. Therefore, whenever we can explicitly compute the inverse of H S (t) we obtain µ S. This is the case of Corollary 3.4 that explains how to obtain µ S when the Hilbert series H S (t) has a certain form. Let B = (b 1,b 2,...,b k ) be a k-tuple of nonzero vectors of Z m such that the semigroup T := b 1,...,b k is pointed and take b Z m. We denote by d B (b) the denumerant, the number of non-negative integer representations of b by b 1,...,b k, that is, the number of solutions of the form b = k i=1 x ib i, where x i is a nonnegative integer for all i. As T is pointed, then d B (b) is finite for all b Z m and d B (0) = 1. It is well known (see, e.g., [13, Theorem ]) that its generating function is given by d B (b)t b 1 =. (1 t b 1 b Z )(1 t b 2) (1 t b k) m Corollary 3.4. Let S be a pointed semigroup such that its Hilbert series is of the form H S (t) = (1 tb 1 )(1 t b 2 ) (1 t b k ) r i=1 c, it d i where b 1,...,b k Z m are nonzero, d 1,...,d r Z m and c 1,...,c r are integers. Then, r µ S (b) = c i.d B (b d i ), for all b Z m, where B = (b 1,...,b k ). i=1 Proof. From Theorem 3.3 and the definition of the denumerant, we have G S (t) = 1 ( r H S (t) = i=1 c it d i r ) = c (1 t b 1 )(1 t b 2) (1 t b k) i t d i d A (b)t b i=1 b Z m = c 1 b Z m d A (b)t m+d1 + +c r b Z m d A (b)t b+dr = c 1 d A (b d 1 )t b + +c r d A (b d r )t b b Z m b Z m = ( r ) c i.d A (b d i ) t b. b Z m i=1 This concludes the proof.

7 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES 7 This result will be particularly important in the following section when trying to obtain explicit formulas for the Möbius function µ S when S is a complete intersection semigroup. Recall that a pointed semigroup S = a 1,...,a n is a complete intersection semigroup if its corresponding toric ideal I S is a complete intersection. Moreover, I S is a complete intersection if it is generated by n d S-homogeneous polynomials, where d is the dimension of the Q-vector space spanned by a 1,...,a n. Whenever I S is a complete intersection generated S-homogeneous polynomials of S-degrees b 1,...,b n d Z m, by [13, Page 341], we have that (4) H S (t) = (1 tb 1 ) (1 t b n d ) (1 t a 1 ) (1 t a n). And so, we will be able to use Corollary 3.4. For characterizations of complete intersection toric ideals we refer the reader to [2, 7]. 4. Explicit formulas for the Möbius function This section is devoted to obtain explicit formulas for the Möbius function µ S when S belongs to some families of subsemigroups of Z m. As far as we are aware, the only known results concerning µ S is an old theorem due to Deddens [5] that determines the values of µ S when S = a,b Z +, and a recent paper due to Chappelon and Ramírez Alfonsín [4] where the authors investigate µ S when S = a,a+d,...,a+kd Z + with a,k,d Z + and obtain a semi-explicit formula when a is even and k = 2. In both papers the authors approach the problem by a thorough study of the intrinsic properties of each semigroup. Here we will first provide in Theorem 4.1 an explicit formula for µ S when S = e 1,...,e m N m ; this formula will be of interest in the last section. After that, we focus on two families of semigroups, namely, the so called semigroups with a unique Betti element and complete intersection numerical semigroups generated by three elements and provide formulas for the Möbius function of these semigroups in Theorems 4.2 and 4.5 respectively. These families generalize the two mentioned above, indeed, if S = a,b Z + then S is a semigroup with a unique Betti element, and if S = a,a+d,a+2d with a even and gcd{a,d} = 1 then S is a complete intersection (see [15, Theorem 3.5]). The results included in this section arise as consequences of Theorem 3.3 (for completeness, we also give a second proof of Theorem 4.5 by means of Theorem 3.2 as an appendix) The semigroup N m. Let {e 1,...,e m } denote the canonical basis of N m. For S = e 1,...,e m = N m the following result holds. Theorem 4.1. For S = N m, the Möbius function is given by ( 1) A if x = i A e i for some A {1,...,m}, µ N m(x) = 0 otherwise. Proof. We observe that H N m(t) = b N m t b = 1 (1 t 1 ) (1 t m ).

8 8 J. CHAPPELON, I. GARCÍA-MARCO, L.P. MONTEJANO, AND J.L. RAMÍREZ ALFONSÍN Therefore, by Theorem 3.3 we derive that G N m(t) = (1 t 1 ) (1 t m ) = and the result follows at once. A {1,...,m}( 1) A i A 4.2. Semigroups with a unique Betti element. t i = A {1,...,m} ( 1) A t i A e i, A semigroup S N m is said to be a semigroup with a unique Betti element b N m if its corresponding toric ideal is generated by a set of S-homogeneous polynomials of S- degree b. These semigroups were studied in detail by García-Sánchez, Ojeda and Rosales in [8]. In particular they prove in [8, Corollary 10] that these are complete intersection semigroups. Theorem 4.2. Let S = a 1,...,a n N m be a semigroup with a unique Betti element b N m and denote by d the dimension of the Q-vector space generated by a 1,...,a n. Then, t ( ) kaj +n d 1 µ S (x) =, j=1 ( 1) A j where {A 1,...,A t } = {A {1,...,n} k A N such that x i A a i = k A b}. Proof. By (4) we have that H S (t) = (1 tb ) n d n i=1 (1 ta i ) = k Aj (1 t b ) n d A {1,...,m} ( 1) A t i A a i. Thus, applying Corollary 3.4, we have that for all x Z m µ S (x) = ( ( 1) A d B x ) a i, i A A {1,...,m} where B is the (n d)-uple (b,...,b). The equality ( k+n d 1 ) if y = kb with k N, k d B (y) = 0 otherwise, for all y Z m, completes the proof. IntheparticularcasewhenS = a 1,...,a n Nisanumericalsemigroupwithaunique Betti element b N, it is proved in [8] that there exist pairwise relatively prime different integers b 1,...,b n 2 such that a i := j i b j, for all i {1,...,n}, and b = n i=1 b i. In this case Theorem 4.2 can be refined as follows. Corollary 4.3. Let S = a 1,...,a n N be a numerical semigroup with a unique Betti element b N. Then, ( 1) A ( ) k+n 2 if x = k i A a i +kb for some A {1,...,n},k N, µ S (x) = 0 otherwise. Proof. As d = 1, by Theorem 4.2 it only suffices to prove that for every A 1,A 2 {1,...,n}, if b divides i A 1 a i i A 2 a i, then A 1 = A 2. For we use [12, Theorem 2.9 and Remark 2.10] (see also [8, Example 12]), which asserts that if we take b 1,...,b n 2 such that a i = j i b j, then the toric ideal associated to S is I S = (f 2,...,f n ), where f i := x b 1 1 x b i i for all i {2,...,n}. Assume that there exist A 1,A 2 {1,...,n},

9 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES 9 A 1 A 2 and that i A 1 a i i A 2 a i = kb for some k N. Thus, the binomial g := i A 1 x i x b 1k 1 i A 2 x i 0 belongs to I S and so it can be written as a combination of f 2,...,f n. However, this is a contradiction because x b i i does not divide j A 1 x j for all i {1,...,n}. As a direct consequence of this result we recover Dedden s classical result. Corollary 4.4. Let a,b Z + be relatively prime integers and consider S := a,b. Then, 1 if x 0 and x 0 or a+b (mod ab), µ S (x) = 1 if x 0 and x a or b (mod ab), 0 otherwise Three generated complete intersection numerical semigroups. We shall study the case when S is a complete intersection numerical semigroup minimally generated by the set {a 1,a 2,a 3 }. In this setting we aim at providing a semi-explicit formula for µ S. This objective is achieved with Theorem 4.5. We include two proofs of Theorem 4.5. In the first one, we prove the theorem as a consequence of Theorem 3.3. In the second one we prove the same result by means of Theorem 3.2, for a sake of brevity we include this second proof as an appendix. Many authors have studied three generated numerical semigroups and one can find several (equivalent) characterizations of the complete intersection property for them. In particular, Herzog proved in [10] that S is a complete intersection if and only if gcd{a i,a j }a k a i,a j with{i,j,k} = {1,2,3};fromnowonweassumethatgcd{a 2,a 3 }a 1 a 2,a 3 and denote d := gcd{a 2,a 3 }. Note that d 2, otherwise a 1 a 2,a 3, which contradicts the minimality of {a 1,a 2,a 3 }. In [10], the author also proved that if we take γ 2,γ 3 N such that da 1 = γ 2 a 2 +γ 3 a 3, then (5) I S = (x d 1 xγ 2 2 xγ 3 3, xa 3/d 2 x a 2/d 3 ), whose S-degrees are da 1 and a 2 a 3 /d, respectively. For every x Z there exists a unique α(x) {0,...,d 1} such that α(x)a 1 x (mod d). It is easy to check that for every x,y Z, α(x) α(y) if α(x) α(y), (6) α(x y) = d+α(x) α(y) otherwise. Theorem 4.5. Let S = a 1,a 2,a 3 be a numerical semigroup such that da 1 a 2,a 3 where d := gcd{a 2,a 3 }. For all x Z, we have if α(x) 2, or µ S (x) = 0 µ S (x) = ( 1) α (d B (x ) d B (x a 2 ) d B (x a 3 )+d B (x a 2 a 3 )) otherwise, where x := x α(x)a 1 and B := (da 1,a 2 a 3 /d). First proof. By (5) I S is generated by two S-homogeneous polynomials whose S-degrees are da 1 and a 2 a 3 /d. Thus, applying (4) we obtain ( )( ) 1 t da 1 1 t a 2 a 3 /d H S (t) = (1 t a 1 )(1 t a 2)(1 t a 3) ( 1 t da 1 )( 1 t a 2 a 3 /d ) = 1 t a 1 t a 2 t a 3 +t a 1 +a 2 +t a 1 +a 3 +t a 2 +a 3 t a 1 +a 2 +a 3.

10 10 J. CHAPPELON, I. GARCÍA-MARCO, L.P. MONTEJANO, AND J.L. RAMÍREZ ALFONSÍN Hence, from Corollary 3.4, we have (7) µ S (x) = d B (x) d B (x a 1 ) d B (x a 2 ) d B (x a 3 )+d B (x (a 1 +a 2 ))+ +d B (x (a 1 +a 3 ))+d B (x (a 2 +a 3 )) d B (x (a 1 +a 2 +a 3 )), for all integers x, where B := (da 1,a 2 a 3 /d). Since α(da 1 ) = α(a 2 a 3 /d) = 0, it follows that α(y) = 0 if y da 1,a 2 a 3 /d. As a consequence of this, d B (y) = 0 whenever α(y) 0. We denote C := {0,a 1,a 2,a 3,a 1 + a 2,a 2 + a 3,a 3 + a 1,a 1 + a 2 + a 3 } and observe that α(y) {0,1} for all y C. We distinguish three different cases upon the value of α := α(x), for x Z. Case 1. α 2. We deduce that α(x y) = α(x) α(y) 0 and d B (x y) = 0, for all y C. Therefore, using (7), µ S (x) = 0. Case 2. α = 1. We deduce that α(x y) 0 and d B (x y) = 0 for all y {0,a 2,a 3,a 2 +a 3 }. Therefore, using (7), we obtain that µ S (x) = d B (x a 1 )+d B (x a 1 a 2 )+d B (x a 1 a 3 ) d B (x a 1 a 2 a 3 ). Case 3. α = 0. Since d 2, we deduce that α(x y) 0 and d B (x y) = 0 for all y {a 1,a 1 +a 2,a 1 + a 3,a 1 +a 2 +a 3 }. Therefore, using (7), we obtain that This completes the proof. µ S (x) = d B (x) d B (x a 2 ) d B (x a 3 )+d B (x a 2 a 3 ). Theorem 4.5 yields an algorithm for computing µ S (x) for all x Z which relies on the computation of four denumerants of the form d B (y), where B = (da 1,a 2 a 3 /d). It is worth to mention that in [16, Section 4.4] there are several results and methods to compute this type of denumerants. Also note that Theorem 4.5 generalizes [4, Theorem 3], where the authors provide a semi-explicit formula for S = 2q,2q+e,2q+2e where q,e Z + and gcd{2q,2q+e,2q+ 2e} = 1. It is easy to see that these semigroups are complete intersection, if suffices to set a 1 := 2q+e, a 2 := 2q, a 3 := 2q+2e and observe that gcd{a 2,a 3 }a 1 = 2a 1 = a 2 +a 3 a 2,a 3. For some other families of complete intersection numerical semigroups we refer the reader to [1, 18]. 5. When is a poset equivalent to a semigroup poset? Let D = {d 1,...,d m } be a finite set and let us consider the (locally finite) poset P of the multisets over D ordered by inclusion. For every S,T P such that T S, it is well known that { ( 1) T\S if T S and T \S is a set, (8) µ P (T,S) = 0 otherwise. We consider the semigroup N m = e 1,...,e m and the map ψ : P N m defined as ψ(s) = (s 1,...,s m ) where s i denotes the multiplicity of d i in S for all S P. If we consider theorderinn m induced by thesemigroup N m we havethat α N m β α i β i for all i {1,...,n} and ψ is an order isomorphism, i.e., an order preserving and order reflecting bijection. Thus, we can say that the poset of multisets of a finite set is a particular case of semigroup poset. This implies in particular that for all S,T P such that T S, then µ P (T,S) = µ N m(ψ(t),ψ(s)) = µ N m(ψ(s) ψ(t)) and, by Theorem 4.1 we retrieve the formula (8).

11 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES 11 It is worth pointing out that we can also use the tools of the previous sections to recover the arithmetic Möbius function, i.e., the Möbius function for the poset of integers ordered by divisibility. Recall that for all a,b N such that a b, then { ( 1) r if b/a is a product of r different prime numbers, (9) µ(a,b) = 0 otherwise. Indeed, for every m Z +, if we denote by p 1,...,p m the first m prime numbers, then one can define an order isomorphism between (N m, N m) and the poset of nonnegative integers that can be written as a product of powers of p 1,...,p m, which we denote by N m ordered by divisibility. Hence for every a,b N we take m Z + such that a,b N m and, hence, we can recover the formula (9) by means of the Möbius function of N m given in Theorem 4.1. In these two examples we have easily found an ad-hoc order isomorphism between the poset and the semigroup poset (N m, N m) which allows us to compute the Möbius function. In this section, we provide tools to do this systematically. Let(P, )bealocallyfiniteposet. Foreveryx P wedenotebyp x := {y P x y}. We aim at studying when the restriction of the Möbius function µ(,x) : P Z can be computed by means of the Möbius function of a pointed semigroup S Z m. Firstly, we observe that µ(,x) is 0 for every y such that x y, then it makes sense to study µ(,x) : P x Z. Of course µ(,x) can be studied by means of the Möbius function of a semigroup S if there exists an order isomorphism ψ : (P x, ) (S, S ). In this section we will characterize in Theorem 5.3 when there exists such an isomorphism in terms of the poset P x. The poset P x is said to be autoequivalent if and only if for all y P x there exists an order isomorphism g y : P x P y such that g y g z = g z g y for all y,z P x and g x is the identity. For all y P x we denote by l 1 (y) := {z P it does not exist u P such that y u z}. Whenever P x is autoequivalent with isomorphisms {g y } x y and l 1 (x) is a finite set of n elements, we associate to P a subgroup L P Z n in the following way. Denote l 1 (x) = {x 1,...,x n } P and consider the map f : N n P defined as f(0,...,0) = x, and for all α N n and all i {1,...,n}, then f(α + e i ) = g xi (f(α)), where {e 1,...,e n } is the canonical basis of N m. In particular, f(e i ) = g xi (f(0)) = g xi (x) = x i for all i {1,...,n}. Lemma 5.1. f is well defined and is surjective. Proof. Suppose that α + e i = β + e j, then we set γ := α e j = β e i N n. Thus, f(α+e i ) = g xi (f(α)) = g xi (g xj (f(γ))) = g xj (g xi (f(γ))) = g xj (f(β)) = f(β +e j ) and f is well defined. Take y P x. If y = x, then y = f(0). If y x, then there exists z P x such that y l 1 (z); therefore y = g z (x j ) for some j {1,...,n}. We claim that if z = f(α), then y = f(α + e j ). Indeed, f(α + e j ) = g xj (f(α)) = g xj (z) = g xj (g z (x)) = g z (g xj (x)) = g z (x j ) = y. Now we set L P := {α β Z n f(α) = f(β)}. Lemma 5.2. L P is a subgroup of Z n. Proof. If γ L P, then γ L P clearly. Moreover, if γ 1,γ 2 L P, then γ 1 + γ 2 L P. Indeed, take α,α,β,β N m such that f(α) = f(α ), γ 1 = α α, f(β) = f(β ) and γ 2 = β β. Then f(α+β) = f(α +β) = f(α +β ) and we are done.

12 12 J. CHAPPELON, I. GARCÍA-MARCO, L.P. MONTEJANO, AND J.L. RAMÍREZ ALFONSÍN For every subgroup L Z n we define the saturation of L as the group Sat(L) := {γ Z n exists d Z + such that dγ L}. Theorem 5.3. Let P be a locally finite poset and x P. Then, (P x, ) is isomorphic to (S, S ) for some (pointed) semigroup S Z n P x is autoequivalent, l 1 (x) is finite and L P = Sat(L P ). Proof. ( ) Let S Z m be a (pointed) semigroup and denote by {a 1,...,a n } its unique minimal set of generators. Assume that ψ : P x S is an order isomorphism; let us prove that P x is autoequivalent, that l 1 (x) = n and that L P = Sat(L P ). Firstly, we observe that setting x i := ψ 1 (a i ), then l 1 (x) = {x 1,...,x n }. Now, for every y P x we set g y : P x P y z ψ 1 (ψ(z)+ψ(y)), then it is straightforward to check that g y is an order isomorphism. Moreover, g x is the identity map in P x and g y g z = g z g y for all y,z P x. Now we take f : N m P x the map associated to {g y } y x, i.e., f(0) = x and if f(α) = y, then f(α + e j ) = g xj (f(α)). We claim that ψ(f(α)) = α i a i S for all α = (α 1,...,α n ) N n. Indeed, ψ(f(0)) = ψ(x) = 0 and if we assume that ψ(f(α)) = αi a i for some α = (α 1,...,α n ) N m, then ψ(f(α+e j )) = ψ(g xj (α)) = ψ(z)+ψ(x j ) = αi a i +a j, as desired. Since L P Sat(L P ) by definition, let us prove that Sat(L P ) L P. We take γ Sat(L P ), then dγ L P for some d Z +. This means that there exist α,β N n such that f(α) = f(β) and dγ = α β. Hence we have that α i a i = ψ(f(α)) = ψ(f(β)) = β i a i ; which implies that γ i a i = 1/d ( (α i β i )a i ) = 0. Thus, if we take α,β N m such that γ = α β, then ψ(f(α )) = ψ(f(β )) and, whence, f(α ) = f(β ) and γ L P. ( ) Since L P = Sat(L P ), we have that Z n /L P is a torsion free group; hence there exists agroupisomorphism ρ : Z n /L P Z m, where m = n rk(l P ). Wedenotea i := ρ(e i +L P ) for all i {1,...,n} and set S := a 1,...,a n Z m. We claim that (P x, ) and (S, S ) are isomorphic. More precisely, it is straightforward to check that the map is an order isomorphism. ψ : P x S y α i a i, if f(α) = y In algebraic terms, the idea under ( ) in Theorem 5.3 is that whenever P x is autoequivalent and l 1 (x) is finite, the subgroup L P defines a lattice ideal I. Moreover, P x is isomorphic to a semigroup poset (S, S ) if and only if the ideal I itself is the toric ideal of a semigroup S, but this happens if and only if I is prime or, equivalently, if L P = Sat(L P ) (see, e.g., [6] or [11]). References [1] I. Bermejo, I. García-Marco, Complete intersections in certain affine and projective monomial curves, Bull. of the Brazilian Math. Society, to appear. [2] I. Bermejo, I. García-Marco, Complete intersections in simplicial toric varieties, arxiv: [3] E. Briales, A. Campillo, C. Marijuán, P. Pisón, Minimal systems of generators for ideals of semigroups. J. Pure Appl. Algebra 124 (1998), no. 1-3, [4] J. Chappelon, J. L. Ramírez Alfonsín, On the Möbius function of the locally finite poset associated with a numerical semigroup, Semigroup Forum 87 (2013), no. 2, [5] J. A. Deddens, A combinatorial identity involving relatively prime integers, J. Combin. Theory Ser. A 26 (1979), no. 2, [6] D. Eisenbud, B. Sturmfels, Binomial ideals, Duke Math. J. 84 (1996) 1 45.

13 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES 13 [7] K. Fischer, W. Morris and J. Shapiro, Affine semigroup rings that are complete intersections, Proc. Amer. Math. Soc. 125 (1997), [8] P. A. García-Sánchez, I. Ojeda and J. C. Rosales, Affine semigroups having a unique Betti element, J. Algebra Appl. 12 (2013), no. 3, , 11 pp. [9] D. H. Greene, D. E. Knuth, Mathematics for the analysis of algorithms. Third edition. Progress in Computer Science and Applied Logic, 1. Birkhäuser Boston, Inc., Boston, MA, viii+132 pp. [10] J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970) [11] A. Katsabekis, M. Morales, A. Thoma, Binomial generation of the radical of a lattice ideal, Journal of Algebra 324, 2010, [12] A. Katsabekis, I. Ojeda, An indispensable classification of monomial curves in A 4 (k), Pacific Journal of Mathematics, to appear; see also arxiv: v2 [math.ac]. [13] M. Kreuzer, L. Robbiano, Computational Commutative Algebra 2. Springer-Verlag Berlin Heidelberg, [14] A. K. Maloo and I. Sengupta, Criterion for complete intersection of certain monomial curves, in: Advances in algebra and geometry (Hyderabad, 2001), Hindustan Book Agency, New Delhi, 2003, pp [15] E. Miller, B. Sturmfels, Combinatorial commutative algebra. Graduate Texts in Mathematics, 227. Springer-Verlag, New York, [16] J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and Its Applications, vol. 30. Oxford University Press, Oxford (2005). [17] J. L. Ramírez Alfonsín, Ø. Rødseth, Numerical semigroups: Apéry sets and Hilbert series, Semigroup Forum 79 (2009), no. 2, [18] J. C. Rosales and M. B. Branco, Three families of free numerical semigroups with arbitrary embedding dimension, Commutative rings, , Nova Sci. Publ., Hauppauge, NY, [19] G-C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, (1964) [20] B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series 8, American Mathematical Society, Providence, RI, Appendix: Second proof of Theorem 4.5 In this appendix we give a second proof of Theorem 4.5 by using Theorem 3.2. Proposition 5.4. Let S = a 1,a 2,a 3 be a numerical semigroup such that da 1 a 2,a 3 where d := gcd{a 2,a 3 }. Then, (a) µ S (x) = µ S (x da 1 )+µ S (x a 2 a 3 /d) µ S (x da 1 a 2 a 3 /d) for all x Z\C, and (b) d 1 d 1 µ S (x ia 1 ) = µ S (x ia 1 a 2 a 3 /d) for all x Z\C, i=0 i=0 where C = {0,a 2,a 3,a 2 +a 3 } and C = C {0,a 1,a 1 +a 2,a 1 +a 3,a 1 +a 2 +a 3 }. Proof. By (5), I S is generated by two S-homogeneous polynomials whose S-degrees are da 1 and a 2 a 3 /d. Applying (4) we get that H S (t) = (1 tda 1 )(1 t a 2a 3 /d ) (1 t a 1 )(1 t a 2)(1 t a 3) = 1 tda1 t a2a3/d +t da1+a2a3/d (1 t a 1 )(1 t a 2)(1 t a 3). A direct application of Theorem 3.2 proves (a). To prove (b) it suffices to observe that (1 t da 1 )/(1 t a 1 ) = 1+t+ +t d 1, giving H S (t) = (1 tda 1 )(1 t a 2a 3 /d ) d 1 i=0 (1 t a 1 )(1 t a 2)(1 t a 3) = (ti t i+(a 2a 3 /d) ) (1 t a 2 )(1 t a 3), and apply again Theorem 3.2. Recall that for every x Z we denote by α(x) the only integer in {0,...,d 1} such that α(x)a 1 x (mod d).

14 14 J. CHAPPELON, I. GARCÍA-MARCO, L.P. MONTEJANO, AND J.L. RAMÍREZ ALFONSÍN Second proof of Theorem 4.5. We set C := {0,a 2,a 3,a 2 + a 3 } and C := C {a 1,a 1 + a 2,a 1 +a 3,a 1 +a 2 +a 3 }. We observe that for all y C, α(y) {0,1}. Moreover α(y) = 0 if y C and α(y) = 1 if y C \C. Let x be an integer, we distinguish three different cases upon the value of α := α(x). Case 1. α 2. We observe that α(x λ(da 1 ) δ(a 2 a 3 /d)) = α(x) 2 for all λ,δ N, which implies that x λ(da 1 ) δ(a 2 a 3 /d) / C. By iteratively applying Proposition 5.4 (a) we get that µ S (x) = 0. Case 2. α = 1. By Proposition 5.4 (b), we have that d 1 d 1 µ S (y ia 1 ) = µ S (y ia 1 a 2 a 3 /d) i=0 i=0 for all y / C. Moreover, for i 2, by Case 1 we have that µ S (x ia 1 ) = µ S (x ia 1 a 2 a 3 /d) = 0. This gives that µ S (y)+µ S (y a 1 ) = µ S (y a 2 a 3 /d)+µ S (y a 1 a 2 a 3 /d) for all y / C. We set σ(y) := µ S (y)+µ S (y a 1 ) and have that (10) σ(y) = σ(y a 2 a 3 /d) for all y / C. Moreover, we observe that µ S (x λa 2 a 3 /d) = 1 for every λ N; hence, x λa 2 a 3 /d / C. Applying (10) iteratively, we get that σ(x) = 0 and µ S (x) = µ S (x a 1 ). Case 3. α = 0. We denote τ(y) := µ S (y) µ S (y da 1 ), then by Proposition 5.4 (a) we have that (11) τ(y) = τ(y a 2 a 3 /d) if y / C. We take λ N the minimum integer such that x λa 2 a 3 /d < 0 or x λa 2 a 3 /d C; an iterative application of (11) yields τ(x) = τ(x λda 1 ), hence µ S (x) = µ S (x da 1 )+ τ(x λda 1 ). Let us compute τ(x λda 1 ). If x := x λda 1 < 0, then τ(x ) = 0. Since α(x ) = α(x) = 0, we have that x C. A direct computation yields τ(0) = 1, τ(a 2 ) = τ(a 3 ) = 1, τ(a 2 + a 3 ) = µ S (da 1 ) µ S (0) = 2 1 = 1 if a 2 + a 3 = da 1 or τ(a 2 +a 3 ) = µ S (a 2 +a 3 ) µ S (a 2 +a 3 da 1 ) = 1 0 = 1 if a 2 +a 3 da 1. Hence, µ S (x) = µ S (x da 1 )+ 1 if x 0 (mod a 2 a 3 /d), 1 if x a 2 (mod a 2 a 3 /d), 1 if x a 3 (mod a 2 a 3 /d), 1 if x a 2 +a 3 (mod a 2 a 3 /d), 0 otherwise. This formula gives µ S (x) in terms of µ S (x da 1 ) and the residue of x modulo a 2 a 3 /d. We iterate this argument to compute µ S (x ida 1 ) for all i 1 such that x ida 1 0. Finally note that for B = (da 1,a 2 a 3 /d), then d B (y) is the number of i N such that y i(da 1 ) 0 and y i(da 1 ) 0 (mod a 2 a 3 /d), hence the result is proved.

15 MÖBIUS FUNCTION OF SEMIGROUP POSETS THROUGH HILBERT SERIES 15 Université Montpellier 2, Institut de Mathématiques et de Modélisation de Montpellier, Case Courrier 051, Place Eugène Bataillon, Montpellier Cedex 05, France address: address: address: address: