Homogeneous numerical semigroups

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen–Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen–Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen–Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.

Keywords Numerical semigroup rings · Tangent cones · Betti numbers

Introduction
Let S be a numerical semigroup minimally generated by a sequence of positive integers n : 0 < n 1 < · · · < n d . For any nonnegative integer j one may consider the shifted sequence n + j : 0 < n 1 + j < · · · < n d + j. Let k be a field and k[S] := k[t n 1 , . . . , t n d ] ⊂ k[t], where t is a free variable, be the numerical semigroup ring defined by S. This is the homogeneous coordinate ring of the affine monomial curve in A d k defined parametrically by x 1 = t n 1 , . . . , x d = t n d . Denote by I (n) ⊂ k[x 1 , . . . , x d ] the defining ideal of k[S] obtained from the natural presentation k[x 1 , . . . , x d ] → k[S] → 0. J. Herzog and H. Srinivasan conjectured that for j 0 the Betti numbers of the ideals I (n + j) become periodic on j with period n d − n 1 . In 2013, the conjecture was proven to be true by Jayanthan and Srinivasan [17] for d = 3, by Marzullo [20] for some particular cases if d = 4, and by Gimenez et al. [10] in the case of arithmetic sequences. Finally, in 2014, Vu gave a completely general positive answer in [35]. One of the main ingredients of Vu's proof is that there exists a positive integer N such that, for all j > N , any minimal binomial non-homogeneous generator of I (n+ j) is of the form x a 1 1 u−vx a d d , where a 1 , a d are positive integers, u, v are monomials in the variables x 2 , . . . , x d−1 , and deg x a 1 1 u > deg vx a d d . It is noteworthy that the bound N depends on the Castelnuovo-Mumford regularity of J (n), the ideal generated by the homogeneous elements in I (n). Let I (n) * be the homogeneous ideal generated by the initial forms of the elements in I (n). Then, k[x 1 , . . . , x d ]/I (n) * G(S), the tangent cone of k[S]. By using this main ingredient in Vu's proof of the conjecture, Herzog and Stamate proved in [15] that for any j > N , the Betti numbers of the ideals I (n + j) and I (n + j) * coincide. Following the general definition given by Herzog et al. [14], we say that S if of homogeneous type if the above condition on the Betti numbers is satisfied. So the result of Herzog-Stamate may be rephrased by saying that S + j is of homogeneous type for any j > N . Note that if a numerical semigroup S is of homogeneous type, then the tangent cone G(S) is Cohen-Macaulay.
In this paper we introduce a new condition on S, to be homogeneous (cf. Definition 3.1), that jointly with the Cohen-Macaulay property of G(S) turns out to be equivalent to a property much similar to the one cited above as the main ingredient of Vu's proof of the Herzog-Srinivasan conjecture (cf. Theorem 3.12). In fact, this property is given in terms of the Apéry set of S and so it can be checked in terms of the generating sequence of integers n. We then show that if S is homogeneous and the tangent cone G(S) is Cohen-Macaulay, then S is of homogeneous type (cf. Theorem 3.17). Taking into account that the Cohen-Macaulay property of G(S) can also be checked in terms of the Apéry set of S, we get a method to prove that a numerical semigroup S is of homogeneous type that only depends on the Apéry set of S and, ultimately, on elementary computations on the sequence n. In addition, we prove that there exists a positive integer L such that for any j > L, all the numerical semigroups generated by sequences of the form n+ j are homogeneous and have Cohen-Macaulay tangent cone (cf. Corollary 6.4), so in particular they are of homogeneous type. The novelty here is that the constant L only depends on the sequence of integers n and can be easily computed. In fact, it can be computed in terms of what we call the shifting type of a numerical semigroup (cf. Definition 6.6): two numerical semigroups can be obtained one from another as a shifting if and only if they have the same shifting type. So our results say that in the class of numerical semigroups with the same shifting type, all numerical semigroups except a finite number, that only depends on its shifting type, are of homogeneous type.
Homogeneous numerical semigroups recover those with a unique maximal expression introduced by Rosales [27]. As a typical example of homogeneous numerical semigroups we have (among several others) those generated by generalized arithmetic sequences (cf. Example 3.6). In this case, the tangent cone is also Cohen-Macaulay and so they are of homogeneous type. For some special cases of the class of numerical semigroups generated by a generalized arithmetic sequence (namely, for d ≤ 4 or n 1 ≤ 2d), this property was proven by Sharifan and Zaare-Nahandi by completely different methods in [33]. On the other hand, numerical semigroups of homogeneous type are not necessarily homogeneous. This is the case for some complete intersection numerical semigroups (cf. Remark 3.16 and Example 3.21). We then explore the difference between both classes of numerical semigroups, and found that in embedding dimension 3 any numerical semigroup which is of homogeneous type has a complete intersection tangent cone or it is homogeneous with Cohen-Macaulay tangent cone (cf. Theorem 4.5). In embedding dimension 4 we give several examples illustrating this difference. In many cases we have checked we get the same conclusion as in embedding dimension 3, so we could ask ourselves if the same is true for larger embedding dimensions, that is, if any numerical semigroup of embedding dimension d ≥ 4 which is of homogeneous type, has a complete intersection tangent cone or it is homogeneous with Cohen-Macaulay tangent cone. But in fact this is not true as we show with a concrete example of embedding dimension 4. By using gluing techniques we also show that for any embedding dimension d, there are infinitely many complete intersection numerical semigroups which are of homogeneous type but not homogeneous (cf. Corollary 5.13). Now, we briefly describe the content of the paper. All the necessary notation and machinery on numerical semigroups is introduced and fixed in Sect. 2. In Sect. 3 we prove our main results on homogeneous numerical semigroups, characterize them, and relate with the property of being of homogeneous type. In Sect. 4 we study in detail the case of embedding dimension 3 and provide different families of examples with embedding dimension 4. Then, in Sect. 5 we study the behavior of the homogeneous property by gluing, particularly for the case of extensions. Finally, in Sect. 6 we study shiftings and prove that the property of being homogeneous and having Cohen-Macaulay tangent cone fulfills asymptotically in the shifting classes. Many of the explicit examples along this paper have been computed by using the NumericalSgps package of GAP [7].
Part of this work has been developed during two stays that the first author has done in the Institute of Mathematics of the University of Barcelona (IMUB) in 2014 and 2016. Both authors would like to thank the IMUB for its hospitality and support. We also thank Anargyros Katsabekis for several discussions on the subject in the case of embedding dimension four, Dumitru Stamate for telling us about his results in [34], and Francesco Strazzanti for providing the example of a numerical semigroup of embedding dimension four which is of homogeneous type but neither homogeneous nor with a complete intersection tangent cone. Finally we would like to thank the anonymous referee for a careful reading of the manuscript and several useful comments and corrections to the paper.

Preliminaries
Let n : n 1 , . . . , n d be a sequence of integers with n 1 < n i for all i = 2, . . . , d, and S = n 1 , . . . , n d be the subsemigoup of (N, +) generated by n. We call S a numerical semigroup when gcd(n 1 , . . . , n d ) = 1 or equivalently N\S is a finite set (cf. [9]). Let k be a field. Then, the sequence n gives rise to a monomial curve C := C(n) ⊆ A d k whose parametrization is given by be the semigroup ring generated by S and set P := k[x 1 , . . . , x d ] the polynomial ring over k. Let I (n) := ker(ϕ), where ϕ : P −→ k[t] is the canonical homomorphism defined by ϕ(x i ) = t n i . We have that P/I (n) k[S] and in the case that S is a numerical semigroup, or k is algebraically closed, then k[S] is the coordinate ring of C(n) and so I (n) is in fact the defining ideal of C(n) (cf. [23]). Moreover, if n is a minimal system of generators of S, the ideal I (n) only depends on S and we set I S := I (n). Let g := gcd(n 1 , . . . , n d ). The numerical semigroup assigned to S is defined as N (S) := n 1 /g, . . . , n d /g .
Note that for any positive integer s, we have s ∈ N (S) if and only if gs ∈ S.
is an isomorphism.
Throughout this paper we consider the natural grading on the polynomial ring. For a vector a = (a 1 , . . . , a d ) of non-negative integers, we use x a to denote the monomial x a 1 1 . . . x a d d . It is known that I (n) is generated by binomials x a − x b where a and b are d-tuples of non-negative integers with ϕ( , is called the support of f and is denoted by supp ( f ). By definition, each element s ∈ S can be written as s = d i=1 a i n i for some non-negative integers a i . The vector a is called a factorization of s and the set of all factorizations of s is denoted by F(s), which is obviously a finite set. Let |a| = d i=1 a i denote the total order of a. Then the maximum integer n which is the total order of a vector in F(s) is called the order of s and is denoted by ord S (s). A vector a ∈ F(s) with |a| = ord S (s), is called a maximal factorization of s and s = d i=1 a i n i is called a maximal expression of s. For a vector a of non-negative integers, we set s(a) = d i=1 a i n i .

Remark 2.2
Let s be an element of S and M = S\{0} be the maximal ideal of S. Then the order of s is the maximum integer n such that nM contains s. In other words, s ∈ nM\(n + 1)M if and only if n = ord S (s).
We use two partial orderings and M on S where, for all elements x and y in S, x y if there is an element z ∈ S such that y = x + z and x M y if y = x + z with ord S (y) = ord S (x) + ord S (z) for some z ∈ S. For a finite subset T ⊂ S, considering these orderings, the maximal elements of T are denoted respectively by Max T and Max M T and the minimal elements of T are respectively denoted by Min T and Min M T . It is clear that Max T ⊆ Max M T and Min T ⊆ Min M T . The following easy fact will be used frequently in our approach. The leading term of a non-zero element f ∈ P is the homogeneous summand of f with least degree, which we denote by f * and set ld( f ) for the degree of f * . For an ideal I ⊂ P we set I * ⊂ P be the graded ideal generated by all In the sequel, we consider the coordinate ring G(S) of the tangent cone of R = k[S], which is precisely the associated graded ring gr m (R) of R with respect to the maximal ideal m = (t n 1 , . . . , t n d ). Note that

Proposition 2.4 G(S) is Cohen-Macaulay if and only if T (S) = ∅.
Let < be a monomial order on P and let f = n i=1 r i x a i be a non-zero polynomial with r i ∈ k. The leading monomial of f with respect to <, denoted by lm < ( f ), is the biggest monomial with respect to < among the monomials {x a 1 , . . . , x a n }. A set of polynomials G = { f 1 , . . . , f n } of an ideal I is called a Gröbner basis of I with respect to <, if {lm < ( f 1 ), . . . , lm < ( f n )} is the set of generators for the monomial ideal lm < (I ) = (lm < ( f ) | f ∈ I ). Since the monomial ideal lm < (I ) has a unique minimal set of monomial generators, the set of leading monomials of elements of any minimal Gröbner basis for I , is a unique set. A Gröbner basis G is called reduced if the coefficient of lm < ( f i ) in f i is one for all 1 ≤ i ≤ n and for i = j, none of the monomials of supp ( f j ) is divisible by lm < ( f i ). Any Gröbner basis of I is a generating set for I , a reduced Gröbner basis exists and it is uniquely determined (cf. [13, Theorem 2.2.7]). We consider the negative degree reverse lexicographical ordering with x 2 > · · · > x d > x 1 . We denote this local term order by < ds , i.e. x b < ds x a precisely when one of the following statements holds:

Remark 2.5 Let
• |b| > |a|; or Remark 2.7 Let f = n i=1 r i x a i be a homogeneous polynomial, where r i ∈ k. Let x a 1 = lm < ds ( f ) and x 1 ∈ supp (x a 1 ). Since x a i < ds x a 1 and |a i | = |a 1 |, we have In particular x 1 divides f .

Semigroups with homogeneous Apéry sets
For an element s ∈ S, the Apéry set of S with respect to s is defined as The Apéry set AP(N (S), t), for t ∈ N (S), is indeed the set of the smallest elements in N (S) in each congruence class modulo t and has t elements. Given 0 = s ∈ S, the set of lengths of s in S is defined as In other words, all expressions of elements in T are maximal. The numerical semigroup S is called homogeneous, when the Apéry set AP(S, n 1 ) is homogeneous.

Example 3.3
A numerical semigroup is called of (almost) maximal embedding dimension if its multiplicity is equal to the embedding dimension (minus one). As in this case, the Apéry set is precisely the minimal set of generators (and one more element of order two), S is homogeneous. [15]), since it is obtained from the semigroup a, b by adding its Frobenius number.
Example 3.5 Recall that I S is called generic if it is generated by binomials with full support i.e. all variables belong to the support of these binomials (cf. [22]). Assume that I S is generic and let E be the minimal set of generators with full support. Let s ∈ AP(S, n i ) with two expressions s = j =i a j n j = j =i b j n j . Then x a −x b ∈ I S should be generated by elements of E. But all elements in E have x i in their support, which is not possible. Hence the elements of AP(S, n i ) have unique expressions, in particular AP(S, n i ) is homogeneous.
Example 3.6 Let S be the numerical semigroup minimally generated by the generalized arithmetic sequence n 0 , n i = hn 0 + it where n 0 , t and h are given positive integers and i = 1, . . . , d. Since we assume that S is a numerical semigroup, the minimal generators are relatively prime and, then, gcd(n 0 , t) = 1. We know from [31] (see also [21]) that Let s = 0 be an element of AP(S, n 0 ). Then, any expression of s cannot involve the generator n 0 , hence if we have an expression of s of length l ≥ 1 it must be of the Because l is a positive integer this implies that l ≥ r 1 d . Let now be any element a = lhn 0 + tr of S with l ≥ r 1 d and let r = qn 0 + r 1 where 0 ≤ r 1 < n 0 . Then w := lhn 0 + tr 1 ∈ S and consequently a = w + tqn 0 does not belong to AP(S, n 0 ) if r ≥ n 0 .
Let 0 = s ∈ AP(S, n 0 ) with two expressions of length l and l . Then Since gcd(n 0 , t) = 1, we have lh = l h + αt and r = r + αn 0 for some integer α. On the other hand r, r < n 0 , since s belongs to AP(S, n 0 ). Hence α = 0 and consequently l = l . Therefore AP(S, n 0 ) is a homogeneous set. Implicitly we have ord S (hn 0 r d + tr) = r d , for all 0 ≤ r < n 0 .

Lemma 3.7 Let c be a factorization of an element s of S. Then there exists a minimal set of generators E of I S such that each binomial x a − x b ∈ E with s(a) / ∈ AP(S, s), has one term divisible by x c .
Proof Let E 1 be a finite set of generators for is again a finite set of generators for I S . Continuing in this way, we get a generating set with the desired property and removing extra elements we have a minimal set of generators.

Lemma 3.8 Let E be a subset of homogeneous binomials in I S and J be the ideal generated by E. Then any binomial in J is homogeneous
.

Proposition 3.9
Let s ∈ S. The following statements are equivalent. (

1) AP(S, s) is homogeneous. (2) For any factorization c of s, there exists a minimal set of generators E for I S such that one term of each non-homogeneous element of E is divisible by x c . (3) There is a factorization c of s, and a minimal set of generators E for I S such that one term of each non-homogeneous element of E is divisible by x c .
Proof (1)⇒(2): Let c be a factorization of s. By Lemma 3.7, there exists a minimal set of generators E of I S such that each binomial . Since all f j are homogeneous by the hypothesis, we get |a| = |b|, from Lemma 3.8. (1) AP(S, n i ) is homogeneous. (

2) There exists a minimal set of generators E for I S such that x i belongs to the support of all non-homogeneous elements of E.
Consider the natural map where π(x 1 ) = 0 and π(x i ) = x i for i = 2, . . . , d. Then For a polynomial f ∈ P, we setf := π( f ) and for a vector of non-negative integers a = (a 1 , . . . , a d ). Finally, we will have a set of minimal generators with the property that, for any x a − x b ∈ E with b 1 = 0, we haveb is a maximal expression.
Next theorem is one of the main results in the paper. (1) S is homogeneous and G(S) is Cohen-Macaulay. (2) For all x a − x b ∈ I S with |a| > |b|, we have s(a) / ∈ AP(S, n 1 ). Moreover ifb is a maximal factorization, then a 1 ≥ b 1 .
(3) There exists a minimal set of binomial generators E for I S such that for all x a − x b ∈ E with |a| > |b|, we have a 1 = 0. (4) There exists a minimal set of binomial generators E for I S which is a standard basis and for all x a − x b ∈ E with |a| > |b|, we have a 1 = 0. (5) There exists a minimal Gröbner basis G for I S with respect to < ds , such that x 1 belongs to the support of all non-homogeneous elements of G and Proof (1)⇒(2): The first statement follows by Definition 3.1. For the second part, let , which implies that s is a torsion element and this contradicts Proposition 2.4.
(2)⇒(3): Let E 1 be a finite set of generators for is again a finite set of generators for I S . Note also that c 1 > a 1 and c 1 > b 1 by the statement of (2). Continuing in this way, we get a generating set with the desired property and removing extra elements we have a minimal set of generators.
Note thatf i is either equal to f i which is homogeneous, or it is a monomial. Therefore B is a homogeneous set of generators for π(I S ). In particular B is a standard basis of π(I S ). Now, using [15, Lemma 1.2] we get that E is a standard basis of I S (see the proof of [15,Theorem 1.4]).
(4)⇒ (5): Note that x 1 is not in the support of homogeneous elements of G from Remark 2.7. Now replacing g i with f j , we get the desired Gröbner basis.
The following example illustrates the fact that even if S is homogeneous and G(S) is Cohen-Macaulay, not any minimal generating set for I S satisfies the properties of the theorem.
} is a minimal generating set (the reduced Gröbner basis) for I S . We can easily see that AP(S, 8) is a homogeneous set, while x 5 2 − x 2 4 is a non-homogeneous element without x 1 in its support. Note that 2 × 25 = 5×10 = 8+3×10+12. Hence replacing } which is also a Gröbner basis and satisfies the properties (3) and (5) of Theorem 3.12.
By a general result due to Robbiano (see [14,24]), Betti numbers of the associated graded ring G(S) are upper bounds for Betti numbers of the semigroup ring R i.e.

Remark 3.15
If S is of homogeneous type, then depth (G(S)) = depth (R) = 1 and so G(S) is Cohen-Macaulay.
Hence the free resolutions of R and G(S) coincide with the Koszul complexes with respect to d − 1 elements and so S is of homogeneous type.
The following result is inspired by the ideas given in the proof of [15,Theorem 1.4].

Theorem 3.17 Let S be a homogeneous numerical semigroup with Cohen-Macaulay tangent cone. Then S is of homogeneous type.
Proof Let E = { f 1 , . . . , f t , g 1 , . . . , g r } be the minimal set of generators of I S which exists by Theorem 3.12(4) and it is also a standard basis . Let f 1 , . . . , f t be homogeneous binomials and g 1 , . . . , g r be non-homogeneous. Then the term of g i which is not the leading term is divisible by x 1 , for i = 1, . . . , r . Hence π(I S ) is a homogeneous ideal and so β i (P/π(I S )) = β i (grn(P/π(I S )), wheren = π(n). Note that G(S) is indeed the completion of gr n (P/I S ) with respect to the m-adic topology. As x 1 is a non-zero-divisor on G(S), it is also a regular element of gr n (P/I S ) and P/I S as well. Hence grn(P/π(I S )) ∼ = gr n (P/I S )/x 1 gr n (P/I S ),  [30], if and only if A admits an artinian reduction (B, n) such that n 2 is principal. Let S be a numerical semigroup. Then k[[S]] is stretched if and only if AP(S, n 1 ) has a unique element of order two. If the order of all elements in the Apéry set is at most two, then it has almost maximal embedding dimension and so it is homogeneous. Assume that AP(S, n 1 ) has an element with order greater than two. Set w ∈ AP(S, n 1 ) the only element of order two. It's easy to see that w = 2n i 1 where n i 1 is a minimal generator of S. Now, let s = d i=2 r i n i ∈ AP(S, n 1 ) with ord S (s) = d i=2 r i > 2. As any subexpression of s is again in the Apéry set, any maximal subexpression of s with length 2 should be equal to w. In particular n i 1 = n i 2 and s = ln i 1 , where l = ord S (s). Therefore {s ∈ AP(S, n 1 ); ord S (s) ≥ 2} has only one maximal element with respect to . Now assume that its Cohen-Macaulay type is equal to d − 1. The Cohen-Macaulay type of S is exactly the number of maximal elements of AP(S, n 1 ) hence Max AP(S, n 1 ) = {tn i 1 , n 2 , . . . , n d }\{n i 1 }, where t is the maximal order of elements of the Apéry set. In particular, S is homogeneous and so it is of homogeneous type if and only if the associated graded ring G(S) is Cohen-Macaulay. This recovers [29,Example 3.5] in the case A is a numerical semigroup ring.
It is proved in [15,Proposition 2.5] that a numerical semigroup generated by an arithmetic sequence is of homogeneous type. For some classes of semigroups generated by generalized arithmetic sequences, it is shown in [33,Corollary 4.12], that they are of homogeneous type. Now, we have: The following example shows that the converse of Theorem 3.17, does not hold even in embedding dimension three.
is minimally generated by a standard basis of two elements. Hence G(S) is complete intersection and so S is of homogeneous type (cf. Remark 3.16), but it is not homogeneous, since 3 × 28 = 4 × 21 = 84 ∈ AP(S, 15).

Small embedding dimensions
We first recall the following definition: a binomial x c i i − j =i x u i j j ∈ I S is called critical with respect to x i if c i is the smallest integer such that c i n i ∈ n 1 , . . . , n i , . . . , n d . The notation c i will be used frequently in the rest of the section. The critical ideal of S, denoted by C S , is the ideal of k[x 1 , . . . , x d ] generated by all critical binomials of I S (cf. [18]).
Recall that a numerical semigroup with Frobenius number F(S), is called irreducible, if it cannot be written as the intersection of two numerical semigroups properly containing it. Let S be an irreducible numerical semigroup. Then S is called symmetric if F(S) is odd and it is called pseudo symmetric if F(S) is even (cf. [9]).
The following classical result by J. Herzog describes the minimal systems of generators of I S for numerical semigroups S with embedding dimension three: (1) If S is symmetric, then after a permutation (i, j, k) of (1, 2, 3), we have (2) If S is not symmetric, then where c i = c ji + c ki for all permutation (i, j, k) of (1, 2, 3) and c i j > 0 for all . As x i , x j belong to the support of all generators of I S , AP(S, n i ) and AP(S, n j ) are homogeneous by Corollary 3.10. Note that c i n i = c j n j . Hence x c i i − x c j j is not a homogeneous binomial. If AP(S, n k ) is homogeneous, then c i n i = c j n j is not in AP(S, n k ). Therefore c i n i = c j n j = r i n i + r j n j + r k n k , for some non-negative integers r i , r j and r k > 0. According to the definition of c i and c j , we get r i = r j = 0 and so r k n k ∈ n i , n j . Let r k = c k + s k . Then c i n i = c j n j = r k n k = c ki n i + c k j n j + s k n k , which implies that c ki = c k j = 0, a contradiction.
(2): By Theorem 4.1, x i , x j , x k belong to the support of all generators of I S and the result follows from Examples 3.5. (2)  We have seen two different classes of numerical semigroups of homogeneous type: homogeneous numerical semigroups which are not symmetric with Cohen-Macaulay tangent cones (cf. Theorem 3.17) and numerical semigroups with complete intersection tangent cones (cf. Remark 4.7). Our next result shows that in embedding dimension three, these two (different) classes determine all numerical semigroups of homogeneous type.

Theorem 4.5 Let S be a numerical semigroup with embedding dimension three. Then the following statements are equivalent. (1) S is of homogeneous type. (2) β 1 (R) = β 1 (G(S)). (3) G(S) is Cohen-Macaulay, and either S is homogeneous or (I S ) * is generated by pure powers of x 2 and x 3 . (4) Either S is non-symmetric homogeneous with Cohen-Macaulay tangent cone, or G(S) is complete intersection.
Proof (1)⇒(2) is clear. Assume that S is not homogeneous. Hence, Theorem 4.2 implies that S is symmetric and k = 1 in the statements of Theorem 4.1. Therefore .

From [32, Corollary 3.2], it follows that
} is a standard basis for I S , (I S ) * = (x c 3 3 , x c 12 2 x c 13 3 , ( f 3 ) * ) and we may assume that c 13 < c 3 . Since c 13 < c 3 , we can not remove neither x c 3 3 nor x c 12 2 x c 13 3 from the set of generators of (I S ) * . On the other hand β 1 Since (x c 3 3 , x c 12 2 x c 13 3 ) is a monomial ideal, if ( f 3 ) * = f 3 , then each monomial term of f 3 belongs to this ideal. Hence x c 2 +c 12 2 ∈ (x c 3 3 , x c 12 2 x c 13 3 ) and consequently c 13 = 0, i.e. (I S ) * = (x c 3 3 , x c 12 2 ). Now, we look at numerical semigroups S with embedding dimension four. We start by observing that, as in the case of embedding dimension 3, S is not necessarily homogeneous neither of homogeneous type. The following example is taken from [5, Remark 3.10]: let S = 16, 18, 21, 27 . Then, S is a complete intersection and G(S) is Gorenstein but G(S) is not a complete intersection, hence the minimal number of generators of the corresponding defining ideals are different and so S is not of homogeneous type. Since G(S) is Cohen-Macaulay, S cannot be homogeneous neither. In fact, 81 ∈ AP(S, 16) and 81 = 3 × 27 = 3 × 18 + 27 are two expressions with different size.
Let S with embedding dimension four. Let C S be the critical ideal of S as defined at the beginning of this section. By a result of A. Katsabekis and I. Ojeda, one can find a minimal system of generators of I S with the following special property:   (2)⇒(3): If c 2 n 2 ∈ AP(S, n 1 ), then c 2 n 2 ∈ n 3 , n 4 . Therefore c 2 n 2 = r 3 n 3 +r 4 n 4 for some non-negative integers r 3 , r 4 . Since n 2 < n 3 , n 4 , we have c 2 > r 3 + r 4 , a contradiction. As n 4 > n 2 , n 3 , a similar argument shows that c 4 n 4 is not in AP(S, n 1 ).

Proposition 4.6 ([18, Proposition 3.9]) Let S be a numerical semigroup of embedding dimension four. Then there exists a minimal system of generators E = E 1 ∪ E 2 of I S , where E 1 is minimal set of generators of C S and E 2 is a set of binomials with full support.
(3) ⇒(1): It follows from Theorem 4.7.
The following well known result by H. Bresinsky provides the systems of generators for the defining ideals of non-complete intersection symmetric numerical semigroups with embedding dimension four. 4 4 − x c 42 2 x c 43 3 , f 5 = x c 43 3 x c 21 1 − x c 32 2 x c 14 Remark 4.10 Let S be a symmetric numerical semigroup of embedding dimension four. If S is not complete intersection, then the set G given in Theorem 4.9, is the unique minimal system of binomial generators for I S (cf. [18,Corollary 3.15]).

Proposition 4.11 Let S be a symmetric and non-complete intersection numerical
semigroup with embedding dimension four. Using the notation of Theorem 4.9, the following statements hold.   8,13,17,15). Since f 4 is homogeneous, by the above proposition S is homogeneous. We also have that G(S) is not Cohen-Macaulay because 15 + 17 = 4 × 8, hence S is not of homogeneous type.
The following two families extracted from [9] can be used to produce symmetric numerical semigroups S with embedding dimension four and given multiplicity m, which are not of homogeneous type neither homogeneous:  Proof In the case m = 2q + 4, we show that I S is generated by First, we check that G 1 ⊆ I S : Hence c 3 = c 4 = 2. From the proof of [9,Lemma 4.22], n 3 + n 4 = (2q + 1)n 2 ∈ AP(S, n 1 ). Therefore c 2 = 2q + 1. Note that n 2 + n 4 − n 3 = m + 2 / ∈ S. So that (q + 1)n 1 = n 2 + n 4 has unique expression, in particular c 1 = q + 1. As the last relation x 1 x 4 − x 2 x 3 is not generated by the others, I S has more than 4 generators and so S is not complete intersection. Now, Theorem 4.9 implies that G 1 is the minimal generating set of I S . Note that none of the critical binomials in G 1 are homogeneous, so it follows that AP(S, n i ) is not homogeneous for all i = 1, . . . , 4, by Theorem 4.7. Now, f := x is not generated by the elements of ( Therefore G 1 is not a standard basis and so β 1 (G(S)) > β 1 (R).
In the second case m = 2q + 5, we show with a similar argument that is the minimal set of generators of I S . Since all critical binomials in G 2 are nonhomogeneous, AP(S, n i ) is not homogeneous for all i = 1, . . . , 4, by Theorem 4.7. Finally, the relation x q+2 2 − x 1 x 3 implies that n 3 is a torsion element and G(S) is not Cohen-Macaulay. In particular, S is not of homogeneous type.
For the pseudo symmetric case, we get the generators of the defining ideal I S from the following result by J. Komeda:  Similarly to the symmetric case, we have the following families of pseudo symmetric numerical semigroups with embedding dimension four and given multiplicity m:    (2), considering q = 1. The defining ideal I S is generated by So this ordering 6, 7, 11, 15 is not the permutation that gives the generators in Theorem 4.16, but S is homogeneous by Corollary 3.10. As 6 + 15 = 3 × 7, G(S) is not Cohen-Macaulay and so S is not of homogeneous type.
In embedding dimension three, we have classified all numerical semigroups of homogeneous type in to numerical semigroups with complete intersection tangent cones and the homogeneous ones with Cohen-Macaulay tangent cones (cf. Theorem 4.5). On other hand, all four generated numerical semigroups of homogeneous type that we have discussed in this section are homogeneous. So it would be natural to look for a similar classification in larger embedding dimensions. Nevertheless, this is not true as the following example with embedding dimension 4 (provided to us by F. Strazzanti) shows: Computing the free resolutions of both S and G(S) by [6], we find their total Betti numbers Therefore S is of homogeneous type and G(S) is not a complete intersection. We also have AP(S, 7) = {0, 8, 11, 12, 16, 20, 24} and that 24 = 8 × 3 = 12 + 12, so S is not homogeneous
In the rest of this section, S will denote the above gluing of S 1 and S 2 .
Definition 5.1 Let S be a gluing of S 1 and S 2 .

Corollary 5.5
If AP(S, qs) is homogeneous for some s ∈ S 1 , then AP(S 1 , s) and AP(S 2 , q) are also homogeneous.
The following example shows that the gluing of two homogeneous numerical semigroups is not necessarily homogeneous.
Example 5.6 Let S := 15, 21, 28 . Then S is an extension of S 1 = 5, 7 with q = 3 and p = 28, but S is not homogeneous from Example 3.21.
The following result is the key for our study of the homogeneity of a gluing: Theorem 5.7 Let S be a gluing of S 1 and S 2 , s ∈ S 1 and n = min{n ∈ N; np / ∈ AP(S 1 , s)}. Then the following statements are equivalent.
(2)⇒(1): Let p, q and a be factorizations of p, q and s, respectively and let E 1 and E 2 be minimal generating sets for I S 1 and I S 2 , respectively, as the ones in Proposition 3.9(2). Now, E = E 1 ∪ E 2 ∪ {x p − y q } is a generating set for I S by Remark 5.3. Note that one term of each non-homogeneous binomial f ∈ E 1 is divisible by x a , and any non-homogeneous binomial g = y c − y b ∈ E 2 has one term divided by y nq . Assume that y c = y nq y d . Then h := x np y d − y b ∈ I S and g is generated by h and x p − y q . Therefore replacing g by h, we get again a generating set for I S . Continuing in this way, we get a generating set for I S which satisfies the property of Proposition 3.9(3), since np / ∈ AP(S 1 , s). Note that, if x p − y q is non-homogeneous, then n = 1 which means that p / ∈ AP(S 1 , s).
As a consequence we obtain the necessary and sufficient conditions for an extension of a homogeneous numerical semigroup to be homogeneous. (1) q = ord S 1 ( p).
(2) p / ∈ AP(S 1 , m 1 ). Now we want to show how to construct systematically non homogeneous numerical semigroups whose tangent cones are complete intersection. The following auxiliary result is needed, where we use the notations and concepts of [1]. Let S = qm 1 , . . . , qm d , pn 1 , . . . , pn k be a nice gluing of S 1 and S 2 . Let G 1 and G 2 be minimal standard bases of I S 1 and I S 2 , respectively. If G(S 1 ) and G(S 2 ) are Cohen-Macaulay, then G = G 1 ∪G 2 ∪{x p −y a 1 } is a minimal standard basis of I S , for some factorization p of p. In particular if G(S 1 ) is complete intersection, then G(S) is also complete intersection.

Lemma 5.10
Proof It follows from the proof of [1, Theorem 2.6]. Proof For d = 3, the result is clear by Lemma 5.11. We proceed by induction on d. Let S 1 = m 1 < · · · < m d−1 be a numerical semigroup of embedding dimension d − 1, with complete intersection tangent cone which is not homogeneous. Let q = 2 and p ∈ S 1 such that ord S 1 ( p) ≥ 2 and gcd(q, p) = 1. Then the extension S := qm 1 , . . . , qm d−1 , p is a nice extension of S. Since S 1 is not homogeneous, S is not homogeneous by Theorem 5.7. More over G(S ) is complete intersection by Lemma 5.10.
As a consequence we get.

Corollary 5.13
Let d ≥ 3. Then there exist infinitely many numerical semigroups with embedding dimension d, which are of homogeneous type but they are not homogeneous.

Shifted family of semigroups
Let n : n 1 < · · · < n d be a sequence of positive integers. For any non-negative integer j, we consider the shifted family n + j : n 1 + j, . . . , n d + j. and the semigroup S + j := n 1 + j, . . . , n d + j , that we call the j-th shifting of S. Remark 6.1 If the semigroup S generated by n is a numerical semigroup, it may happen that S + j is not anymore a numerical semigroup. For instance, let S = 4, 7 . Then S + 2 = 6, 9 . Also, it may happen that n is a minimal system of generators of S but the shifted family is not anymore a minimal system of generators of S + j. For instance, S = 5, 11, 13 and S + 1 = 6, 12, 14 = 6, 14 .
Lemma 6.2 If S is the numerical semigroup minimally generated by n, then S + j is minimally generated by n + j for all j > n d − 2n 1 .
Proof Assume that n r + j = d i=1 s i (n i + j) for some non-negative integers s i . Let a = d i=1 s i . If a ≥ 2, then n r + j ≥ a(n 1 + j) ≥ 2n 1 + 2 j > n d + j.
Hence n r > n d , a contradiction.
We will use the following notation in the sequel: is the Castelnuovo-Mumford regularity of J (n). It is shown by Vu, in [35,Corollary 3.6], that for any j > N , for inhomogeneous prime binomials x a − x b of I S , x 1 divides x a where |a| > |b|. Using this fact, Herzog and Stamate [15,Theorem 1.4] show that S + j is of homogeneous type, in particular G(S + j) is Cohen-Macaulay, for all j > N . In the following result, we improve this bound N by L, which only depends on the initial data of the family n, using ideas inspired by [35].
In particular a 1 = 0 and the result follows. Now assume that d > 2. Let x := d i=1 a i m i and c be a factorization of x with |c| = l, in the semigroup m 1 , . . . , m d−1 , whose support has the largest cardinality among all factorizations of x with total order l. In our argument we use several times the fact that m 1 > m 2 > · · · > m d−1 > m d = 0. We show, arguing by contradiction, that c 1 = 0. If c 1 = 0, then x ≤ lm 2 and so x/m 2 ≤ l.
In particular the second inequality holds by (2), since m d = 0, and the last one follows because n d + j > L + n d > gc + dm 1 . As y > gc + m 1 , we may write y = tm 1 + v for some integers t > 0 and gc ≤ v < gc + m 1 . Note that both y and m 1 are divisible by g and so g|v.
Since v/g ≥ c, v/g ∈ T and so v/g = d−1 i=1 w i (m i /g) for some w i ≥ 0. Hence As v < gc + m 1 Note that n d + j > n d + L = m 1 m 2 gc+m 1 m d−1 + d , therefore Now, we have On the other hand, d−1 i=1 m i + gc + m 1 < n d + j ≤ x. Therefore we may write x = d i=1 z i m i , where z 1 = t + w 1 , z i = δ i,F + w i for i = 2, . . . , d − 1 and z d = l − ( d−1 i=1 w i + t + |F|). Now, z is a factorization of x with larger support than c, a contradiction. Therefore c 1 = 0. Note that s = l(n d + j) − x = d i=1 c i n i . Hence we can take b = c and the result follows.

Corollary 6.4 If j > L, then S + j is homogeneous and G(S) is Cohen-Macaulay.
Proof It follows by Theorem 3.12(3). Corollary 6.5 [15] If j > L, then S + j is of homogeneous type.
Proof It follows by Corollary 6.4 and Theorem 3.17. Definition 6.6 Given a sequence of positive integers s : s 0 = 0 < s 1 < · · · < s d−1 , d ≥ 2, we say that n is of shifting type s if s i = n d − n d−i , for all 1 ≤ · · · ≤ d − 1. We also say that the semigroup S = n 1 , . . . , n d is of shifting type s. Remark 6.7 Note that n d−i = s d−1 − s i + n 1 for all i = 1, . . . , d, hence the sequence n is uniquely determined by n 1 and its shifting type. We may reformulate the above results in the following way: Proposition 6.9 Given a sequence of positive integers s : s 0 = 0 < s 1 < · · · < s d−1 , for any e > L all the numerical semigroups S = n 1 , . . . , n d with n 1 = e and shifting type s are homogeneous and G(S) is Cohen-Macaulay.
By using the notation in [15], we define the width of a numerical semigroup S as the difference between the largest and the smallest generator in a minimal set of generators of S, and denote this number as wd(S). It is clear that for a given positive integer w ≥ 2, there is only a finite number of possible sequences s for the shifting type of the numerical semigroups whose width is bounded by w. So we finally conclude that: Proposition 6.10 Let w ≥ 2. Then, there exists a positive integer W such that all the numerical semigroups S with wd(S) ≤ w and multiplicity e ≥ W , are homogeneous and G(S) is Cohen-Macaulay. Example 6.11 Let a < b be positive integers. Then S k = k, k + a, k + b is a numerical semigroup with shifting type s 1 = b − a, s 2 = b, for any k > 0 and

Remark 6.12
For numerical semigroups S k = k, k + a, k + b of embedding dimension three, the given bound by Vu [35], is improved by Stamate in [34,Theorem 3.5] showing that the Betti numbers of S k are periodic in k, for k > k a,b = max{b( b−a g − 1), b a g }. Moreover, S k is of homogeneous type for k > k a,b . As the above example shows, in embedding dimension three, k a,b is a better bound.