The tree of good semigroups in $\mathbb{N}^2$ and a generalization of the Wilf conjecture

In this work, we study good semigroups of $\mathbb{N}^n$ introducing the definition of length and genus for these objects. We show how to count the local good semigroup with a fixed genus. Furthermore, we study the relationships of these concepts with other ones previously defined in the case of good semigroups with two branches.


Introduction
The study of good semigroups was formerly motivated by the fact that they are the value semigroups of one-dimensional analytically unramified rings (such as the local rings of an algebraic curve). The definition appeared the first time in [1] and these objects were widely studied in several works [2,5,10,11,18]. In [1], the authors proved that the class of good semigroups is actually larger than the one of value semigroups. Thus, such semigroups can be seen as a natural generalization of numerical semigroups and studied using a more combinatorial approach without necessarily referring to the ring theory context. In recent works [9], [20], [8], some of notable elements and properties of numerical semigroups has been generalized to the case of good semigroups. The main purpose of this work is to generalize the definitions of length and genus of an ideal of a numerical semigroup to the case of good ideals of a good semigroup by identifying the relationships between them and those already defined in the previous works in the case of subsemigroups of N 2 . If R is an analytically unramified ring, the value semigroup v(R) = {v(r) | r is not a zerodivisor of R} is a good semigroup [7]. If I is a relative good ideal of R, the extension I ⊆Ī is of finite type and the conductor ideal is C(I) = I :Ī, where both the closure and the colon operation are considered in the ring of total fractions. Fixed α ∈ Z d , we denote by I(α) = {r ∈ R | v(r) ≥ α}. If c(v(I)) is the conductor of the ideal v(I) of v(R), we have v(I :Ī) = v(I(c(I))). In the one-branch case, given a relative ideal I of R, we have that the length of the R-module l R (I/C(I)) = n(v(I)), where n(v(I)) is the cardinality of the set of small elements of the numerical semigroup v(I). For this reason, given a relative ideal E of a numerical semigroup S, it is natural to call length of E the number n(E). On the other hand, the genus of E is defined as the number of gaps in E and it is denoted by g(E). It is straightforward that g(E) + n(E) = c(E). In Section 1 we recall the definition of good semigroup and we fix the basic notations. In [7] it is defined a function of distance d between relative good ideals in a good semigroup S and it is proved that if S is a semigroup of values of a ring R, given a good relative ideal I of R, we have l R (I/C(I)) = d(v(I) \ v(C(I))). For this reason, taking in account the additivity of function d, we can generalize in a natural way the definition of length and genus to the case of good ideals of N d (not necessarily in case of semigroup of values of a ring) as it was done in [26] for Arf semigroups. Given a relative good ideal E ⊆ S, we define respectively length of E and genus of E as l(E) = d(E \ E(c(E))) and g(E) = d(N d \ E(c(E)). We conclude the section giving a slightly different version of an explicit method to compute length and genus introduced in [11]. In [26], it was computed the number of good Arf semigroups with n branches having a fixed genus using the untwisted multiplicity trees defined in [25]. Our aim is to obtain a similar result for a general good semigroup. In [3], it is presented a method to compute all numerical semigroups up to a fixed genus building a tree where each new level is obtained removing minimal generators larger than the Frobenius number from the semigroups of the previous level. In Section 2, we repeat the same idea for good semigroup S ⊆ N 2 ; in this case, the tracks of the good semigroup, defined in [20], will have the role that minimal generators played in case of numerical one. In order to do this, we prove that every good semigroup of genus g can be obtained removing a track from one of genus g − 1 (Theorem 2.4) and that by removing a track from a good semigroup of genus g we obtain a good semigroup of genus g + 1 (Theorem 2.5). Then, we explain how to build the tree of good semigroups, underlining the differences with the numerical case. We report the results regarding the computation of the number of local good semigroup with a fixed genus until genus 27, produced with an algorithm written in [17] using the package [12]. In Section 3 we study the relationship of the genus and the length with others notable elements of a good semigroup. In [15], it was proved the inequality c(S) ≤ (t(S) + 1)l(S) for numerical semigroups. The type of a good semigroup was originally defined in [1] for the good semigroups such that S − M is a good relative ideal of S and it was recently generalized in [8]. We conclude the paper asking if this inequality holds for good semigroup with respect the generalized version of definition of type. We definitely prove that length, genus and type satisfy the relationships t(S) + l(S) − 1 ≤ g(S) ≤ t(S)l(S) also in the case of good semigroups (Proposition 3.6 and Corollary 3.7). To conclude the paper we observe that the definitions and the algorithm given in the previous section give us the possibility of introducing an analogous of the Wilf conjecture for good semigroups. In this case, we found counterexamples for the conjecture (Example 3.8) but the problem seems to stay open for good semigroups which are value semigroups of a ring.

Length and Genus of a Good Ideal
We begin recalling the definition of good semigroup introduced in [1].
is a good semigroup if it satisfies the following properties: Furthermore, we always suppose to work with a local good semigroup S, that is, if α = (α 1 , . . . , α d ) ∈ S and α i = 0 for some i ∈ {1, . . . , d}, then α = 0. As a consequence of property (G2), the element c(S) = min{δ|S ⊇ δ + N d } is well defined and it is called conductor of the good semigroup. The element f (S) = c(S) − 1, where 1 = (1, . . . , 1), is said Frobenius vector of the good semigroup. Furthermore, we denote by Notice that the good semigroup with small elements {(0, 0), (1, 1)} is a local good semigroup containing all the other ones. We denote it by N 2 (1, 1). Following the notations reported in [7], we recall some definitions. Let S be a good semigroup. If E ⊆ Z d is such that E + S ⊆ E and α + E ⊆ S for some α ∈ S, then E is called a relative ideal of S. A relative ideal of S need not satisfy the properties (G1) and (G3) of good semigroups. A relative ideal E that does satisfy properties (G1) and (G2) will be called a good relative ideal. Given a good semigroup S and an its relative good ideal E, two elements α, β in E are said consecutive if there are no elements γ ∈ E such that α < γ < β. An ordered sequence of n + 1 elements in E: is said a chain of length n in E; furthermore it is called saturated in E if all its elements are consecutive in E. In [7] it is proved that, if α, β ∈ E with α < β, then all the saturated chains between α and β have the same length. This common length is denoted by d E (α, β).
Given two relative good ideals E ⊇ F , denoting by e E , e F the minimal elements of E and F respectively, we can define: in [7] it is proved that, providing to take α sufficiently large, this definition is independent of the choice of α. Given α ∈ N d , we denote by E(α) = {β ∈ E : β ≥ α}. The function d(− \ −) satisfies the following properties: 3) Let us consider E, a good relative ideal of S and α ∈ Z d . If Given a good ideal E, there always exists a minimal element  Now we want to introduce an useful formula for the computation of the genus. As in [6] and in [20], we will work on the equivalent structure of semiring Γ S in order to simplify the notation. We set N = N ∪{∞} and extend the natural order and the sum over N to N, setting respectively, a < ∞ for all a ∈ N and x + ∞ = ∞ + x = ∞.
The proof of the second formula is analogous to the first one.
We denote by A f (S) the set of finite absolutes in S (these are also called maximal elements for example in [1], [11]). Furthermore, we denote by A ∞ (S) the set of infinite absolutes in S and with A(S) the set of all absolutes in S. We call I A (S) the set of all irreducible absolutes in S.
It is easy to see that the set of irreducible absolutes is finite and in [20] it is proved that S is generated by these elements as a semiring.
Proof. In this case, by the last proposition, follows g(S) = g( ). We notice that: and observing that the second set has the same cardinality of A f (S), we obtain the thesis.

The tree of good semigroups of N by genus
In [3], it is presented a method to compute all numerical semigroups until a fixed genus building a tree where each new level is obtained removing minimal generators larger than the Frobenius number from the semigroups of the previous level.
In this section we want to repeat the same process with the good semigroups, building a tree of local good semigroups of N 2 , where the gth level of the tree collects all local good semigroups with genus g. In order to follow this idea, we will show that, in this case, the analogous of minimal generators are the tracks of the good semigroup originally defined in [20].
We will simply say that T ⊆ S is a track in S if there exist α 1 , . . . , α n ∈ I A (S) such that T is the track connecting α 1 , . . . , α n . Notice that the previous definition implies that a track T of S never contains elements α such that α ≥ c(S) + e(S).
Given A, B ⊆ N d we set: In the next two theorems we will establish a relationship between the tracks and the genus of a good semigroup.
can be obtained removing a track from a good semigroup with genus g(S) − 1.
Case 1: f (S) / ∈ S. In this case we introduce the following sets: Notice that S = N 2 (1, 1) implies that at least one between X and Y can be always considered. If X and Y are defined and not empty we consider respectively:x = max{x | (x, f 2 ) ∈ S} and y = max{y | (f 1 , y) ∈ S}. If X = ∅, we denote by otherwise we denote by If Y = ∅, we denote by otherwise we denote by We consider the following sets: In order to prove the thesis we can reduce to consider only the good semigroup S 2 . According to definition of tracks, it is easy to observe that T X is a track of S 2 having the form . As a consequence of Corollary 1.8 we have that S is obtained removing a track by the good semigroup S 2 having genus g(S) − 1. If X = ∅, removing the track T X , we have that |A f (S 2 )| = |A f (S)| + 1, g(S 2 2 ) = g(S 2 ) and g(S 2 1 ) = g(S 1 ). Hence, using again Corollary 1.8 we have that S is obtained removing a track by the good semigroup S 2 having genus g(S) − 1. also in this case it easy to observe that S ′ is a good semigroup and T is a track of S ′ . Removing the track T X from S ′ , we have that |A f (S ′ )| = |A f (S)| + 1, g(S ′ 2 ) = g(S 2 ) and g(S ′ 1 ) = g(S 1 ). Hence, using again Corollary 1.8 we have that S is obtained removing a track by the good semigroup S ′ having genus g(S) − 1. Proof. Let be S a good semigroup, we suppose to consider a track T := T ((α 1 , . . . , α n )).
With the same notation used in Lemma 2.3, we consider S ′ = S \ T . We want to prove that g(S ′ ) = g(S)+1. We observe that by the Definition 2.1, α i ∈ A f (S), for any i ∈ {2, . . . , n−1}.
In order to fix the notation, we will suppose α 1 = (x, ∞) and we distinguish two cases. Case 1: α n ∈ A f (S). In this situation, removing the track T , we remove the elements α i with i = 1, . . . , n, hence we lose n − 1 finite absolutes, in S ′ . At the same time the elements γ i = min{α i , α i+1 } ∈ S ′ with i = 1, . . . , n − 1 become finite absolutes of S ′ ; therefore |A f (S)| = |A f (S ′ )|. Since α i,2 = γ i−1,2 , for any i ∈ {2, . . . , n}, we have also g(S ′ 2 ) = g(S 2 ). Now we observe that ∆ S ′ 1 (α n ) = ∅, since S ′ ⊂ S and α n was an absolute of S, furthermore 1 ∆ S ′ (α n ) = ∅, from the definition of S ′ . Hence in S ′ we lose an element in the first projection; we have g(S ′ 1 ) = g(S 1 ) + 1. From Corollary 1.8: y). In this situation, removing the track T , we remove the elements α i with i = 1, . . . , n, so in S ′ we lose n − 2 finite absolutes in S ′ . At the same time, the elements Since α i,2 = γ i−1,2 , for any i ∈ {2, . . . , n} and α i,1 = γ i,1 , for any i ∈ {1, . . . , n − 1}; we have respectively g(S ′ 2 ) = g(S 2 ) and g(S ′ 1 ) = g(S 1 ). Therefore, again from Corollary 1.8: Now we suppose α 1 ∈ A f (S); in this case, if α n = (∞, y) the proof of the Theorem is analogous at the Case 2 that we have seen above, so we can suppose α n ∈ A f (S). In this case removing the track T , we remove the elements α i , i = 1, . . . , n, so we lose n finite absolutes in S ′ . At the same time the elements of γ i i = 1, . . . , n − 1 become finite absolutes in S ′ ; hence |A f (S)| = |A f (S ′ )| + 1. With the same argument that we used in Case 1, it is easy to observe that g(S ′ 1 ) = g(S 1 ) + 1 and g(S ′ 2 ) = g(S 2 ) + 1. So we have: We have observed that tracks play the same role of minimal generators in the case of numerical semigroups. Now we want to show that, as in the numerical case, it is not necessary to consider all the minimal generators in order to build the tree. In fact, as we are going to show, it is sufficient to consider some special tracks.
Definition 2.6. Given a local good semigroup S ⊆ N 2 , we say that a track T ((α 1 , . . . , α n )) is a beyond track, if {α ∈ S, α ≥ c(S)} ∩ T ((α 1 , . . . , α n )) = ∅. Furthermore, we denote by If S is a good semigroup of N 2 obtained from a good semigroup S ′ removing a track, we say that S ′ is a parent of S (or equivalently S is a son of S ′ ). We say that S ′ is a special parent of S (or equivalently S is a special son of S ′ ), if S is obtained from S ′ removing a beyond track.
Lemma 2.7. If S ′ is a special parent of S, then c(S ′ ) < c(S).
Proof. Since S ′ is a special parent of S, we have S = S ′ \ T , where T is a beyond track of S ′ . In particular we have S ⊆ S ′ , implying that c(S ′ ) ≤ c(S). Thus, we need to prove that c(S ′ ) = c(S). Let us assume by contradiction that c(S ′ ) = c(S). Since T is a beyond track of S ′ , there exists an element β ∈ T with β ≥ c(S ′ ). We recall that for a good semigroup S we have that α ∈ S ⇐⇒ min(α, c(S)) ∈ S. Thus, since β / ∈ S, we have that is a contradiction.
From Theorem 2.4 we can deduce the following corollary. 1) If f (S) ∈ S, then S has exactly one special parent.
2) If f (S) / ∈ S, then S has exactly p special parents.
Proof. 1) If we define the set T as in the proof of Theorem 2.4, the good semigroup S ′ is trivially a special parent of S. Now we want to prove that, if there exists a good semigroup S ′ and a beyond track T ′ of S ′ , such that S = S ′ \ T ′ , then T ′ = T . If f (S) ∈ S then, by the previous lemma, there exists β ∈ ∆(f (S)) ∩ T ′ . By property (G3) of good semigroups, it follows ∆(f (S)) ⊆ T ′ . In this case, since (f 1 , ∞) and (∞, f 2 ), are respectively a point of start and a point of end, we have T ′ = ∆(f (S)).
2) If we define, when it is possible, the sets T X and T Y as in the proof of Theorem 2.4, the good semigroups S 1 and S 2 are trivially special parents of S. Now we want to prove that, if S ′ is a good semigroup and there exists a beyond track T ′ of S ′ such that S = S ′ \ T ′ , then T ′ = T X or T ′ = T Y . By the previous lemma, there exists β ∈ ∆(f (S)) ∩ T ′ . Thus, by property (G3) of good semigroups, it means either ∆ 1 (f (S)) ⊆ T ′ or ∆ 2 (f (S)) ⊆ T ′ . We suppose ∆ 1 (f (S)) ⊆ T ′ , and we prove that T ′ = T Y . We start observing that (f 1 , ∞) is always a point of start. If the set Y is empty, then it is also a point of end, hence . Therefore by the maximality ofỹ, it follows y =ỹ; so we have T ′ = T ((f 1 , ∞), (∞,ỹ)) = T Y . Assuming ∆ S 2 (f (S)) ⊆ T ′ , repeating the same proof, it is easy to observe that T ′ = T X . Now, we denote by N g the set of good semigroups with genus g. We build the following family of sets of good semigroup: We want to show that all semigroups of genus g + 1 are special sons of semigroups of genus g, in other words: Proposition 2.9. A g = N g , for any g ∈ N \ {0}.
Proof. We work by induction on g. We suppose A g−1 = N g−1 and trivially A g ⊆ N g . If S ∈ N g , in Proposition 2.8 we have proved that there exists a special parent S ′ that, by Theorem 2.5, belongs to N g−1 = A g−1 ; so S ∈ A g .
So we can build all the semigroups of genus g by removing special tracks from semigroups of genus g − 1.
Remark 2.10. We denote by n g the number of local good semigroups of genus g. The fact that a good semigroup of N g can have two distinct special parents in N g−1 implies that the formula does not hold in general. From the computational point of view, it would be convenient to determine, for each good semigroup S, a subset T (S) ⊆ BT(S) such that In order to do that we define, given a good semigroup S with conductor c(S) = (c 1 , c 2 ), and we claim that T (S) satisfies the required property. In order to do that it suffices to show that each good semigroup S has one and only one special parent S ′ , such that S = S ′ \ T with T ∈ T (S ′ ). Thus, let us consider an arbitrary good semigroup S, with conductor c(S) = (c 1 , c 2 ). Case 1: c 2 = 1. Corollary 2.8 tells us that S has only one special parent S ′ , such that S = S ′ \T , with T ∈ BT(S ′ ). Furthermore, denoting by c(S ′ ) = (c ′ 1 , c ′ 2 ), we still have c ′ 2 = 1. Hence, by definition of T (S ′ ), we have BT(S ′ ) = T (S ′ ) and T ∈ T (S ′ ) as required. Case 2: c 2 = 1 and f (S) = (f 1 , f 2 ) ∈ S. Corollary 2.8 tells us that S has only one special parent S ′ , namely S ′ = S∪∆(f (S)). Notice that ∆(f (S)) is the track T = T ((f 1 , ∞), (∞, f 2 )) ∈ BT(S ′ ). Since we have c(S ′ ) ≤ f (S), it follows that (∞, f 2 ) ≥ (∞, c ′ 2 ), implying that T ∈ T (S ′ ). Case 3: c 2 = 1 and f (S) = (f 1 , f 2 ) / ∈ S. In this case we have exactly two special parents Notice that T Y never belongs to T (S 1 ), since c(S 1 ) = (c 1 − m, c 2 ) and (∞,ỹ) < (∞, c 2 ). On the other hand, T X always belongs to T (S 2 ) because, in both the possible definitions, the end point is (∞, f 2 ) ≥ (∞, c(S 2 ) 2 ) (we recall that c(S 2 ) has the form (c 1 , c 2 − n)). Thus S 2 is the required unique special parent of S. Now we show with an example the construction of the tree up to genus g = 4.
3 Relationship between genus and other notable elements 3.1 On the type of a good semigroup In this subsection we want to relate the genus and the type of a good semigroup S ⊆ N 2 by generalizing a well known inequality that holds in the case of numerical semigroups. First of all, we recall the concept of type of a good semigroup by following the definition introduced in [8] that extends the one initially given in [1]. We write that (α 1 , α 2 ) ≤≤ (β 1 , β 2 ) if and only if (α 1 , α 2 ) = (β 1 , β 2 ) or (α 1 , α 2 ) = (β 1 , β 2 ) and in the same way we write (α 1 , α 2 ) ≪ (β 1 , β 2 ) if and only if α 1 < β 1 and α 2 < β 2 . Given a good semigroup S ⊆ N 2 , let us consider a set A ⊆ S such that there exists c ∈ N 2 with c + N 2 ⊆ S \ A. As described in [8], it is possible to build up a partition of such set A, in the following way. Let us define, D (0) = ∅: For a certain N ∈ N, we have A = ∪ N i=1 D (i) and D (i) ∩ D (j) = ∅. In according to notation of [8], we rename these sets in an increasing order setting A i = D (N +1−i) . Thus we have and the A i are called levels of A. It was proved [Thm. 2.7 [8]] that, if E = S \ A is a proper good ideal of S, then N = d(S \ E).
Definition 3.1. Let us consider a set A ⊆ N 2 such that there exists c ∈ N 2 with c+N 2 ⊆ N 2 \A.

We denote by NL(A) the integer such that
is the partition in levels of A described above.
Now we want to generalize to good ideals a result proved for good semigroups in [8]  Proof. If I is a good relative ideal of S there exists α ∈ S such that α + I ⊆ S. We notice that (α + I) ∪ {0} is a good semigroup. In fact, if α + i 1 , α + i 2 ∈ α + I, so we can write: We can conclude that: Using this result we can rewrite easily the genus as the number of level of N 2 \ S. Proof. It is sufficient to apply Proposition 3.2, considering S = T , I = N 2 , A = N 2 \ T .
In [8], the set of pseudo-frobenius elements of a good semigroup S is defined as The set PF(S) satisfies the condition of the set A in Definition 3.1. The type of the good semigroup S is defined as t(S) = NL(PF(S)), that is the number of levels of the pseudofrobenius elements.
Remark 3.4. We recall that, given two ideals E and F of S, it is possible to consider the set This set is not in general a good ideal. We have that PF(S) = (S − M) \ S, thus t(S) = NL((S − M) \ S). In [8,Proposition 3.5], it is proved that if S − M is a good ideal, then t(S) = d(S − M \ S) as it was initially defined in [1].
Given a good semigroup S, we want to prove the inequality g(S) ≤ t(S)l(S), that generalizes the analogous one proved in [15] for numerical semigroups.
In order to do that we need some lemmas.
Lemma 3.5. Let us consider a subset A ⊆ N 2 such that there exists c ∈ N 2 with c+N 2 ⊆ N 2 \A.
Proof. Denote by n = NL(A) and by m j = NL(B j ). Furthermore we write We want to find a chain and such that each B j,l contains at most one of the α i 's. In order to do that we consider α 1 as an arbitrary element of A 1 , then we choose α i , with i ≥ 2, by taking in account the following rule: Case 1: Denote by D = A i+1 ∩ {β ∈ A|α i ≪ β}. If D is not empty, then we choose as α i+1 an arbitrary element of D. Case 2: If D = ∅, then [8, Lemma 2.4 (1)] ensures that ∆ 1 (α i ) ∩ A i+1 and ∆ 2 (α i ) ∩ A i+1 are both non-empty. Furthermore, if we suppose that α i ∈ B j,l , then there must exist a k ∈ {1, 2} such that ∆ k (α i ) ∩ B j,l = ∅. In fact, otherwise, we would have α i = β 1 ⊕ β 2 , with α i , β 1 , β 2 ∈ B j,l that it is a contradiction since B j,l is a level of B j and cannot contain such a configuration. Thus, in this case we choose an element of ∆ k (α i ) ∩ A i+1 as α i+1 . By construction and by the properties of the levels, it is clear that it is not possible to find l ∈ {1, . . . , m j } and j ∈ {1, . . . , h} such that |B j,l ∩ {α 1 , . . . , α n }| ≥ 2.
Thus NL(A) ≤ h j=1 NL(B j ) as required. Now we are ready to prove the main result of this subsection. Proof. Denote by n = l(S). We choose an arbitrary saturated chain in S between 0 and c(S). We consider the following chain of ideals of S: We have and by Lemma 3.5 and Corollary 3.3 we can deduce: Now we claim that for all i = 2, . . . , n. For each i ∈ {2, . . . , n}, we denote bỹ , and we consider the following function The function f is clearly injective, thus in order to prove our claim we need only to show that it is well defined. Thus we fix an arbitrary β ∈ (S − S(α i )) \ (S − S(α i−1 )) and we prove that β +α i−1 ∈ (S − M) \ S.

On the Wilf Conjecture
In [23], it was firstly introduced the well known Wilf conjecture regarding the numerical semigroups. In [13] it was rephrased in the context of numerical semigroups. It states that the number of minimal generators of a numerical semigroup S, i.e. the embedding dimension of the semigroup, always satisfies the inequality where c(S) and g(S) are respectively the conductor and the genus of the semigroup. The Wilf conjecture represents an important open problem in the context of the numerical semigroup theory, and it has been proved for many special cases [15], [19], [22], [21], [14], [4] and checked for numerical semigroups up to genus 50 in ( [3]) and up genus 67 in [16].
In [20] the concept of embedding dimension of a good semigroup of N 2 is introduced, therefore it makes sense to extend in a natural way the conjecture to these more general objects.
Specifically we want to check if, for a good semigroup S ⊆ N d of genus g, the inequality always holds, where c S is defined as in Section 1. By exploring the tree of good semigroups of N 2 , introduced in the previous section, it is possible to check that the conjecture is satisfied for semigroups up to genus 22. However, starting from genus 23, examples where the conjecture is not verified begin to show up, as it is shown in the following example.
We have c S = 34 and g(S) = 23. Using the algorithms presented in [20], it is possible to check that edim(S) = 3 (according to the terminology introduced in that paper, the set {(4, 3), (8, ∞), (13, 11)} ⊆ I A (S) constitutes a minimal set of representatives for S). Remark 3.9. It still makes sense to ask whether the Wilf conjecture is true for good semigroups that are value semigroups. In fact, at the moment, there are no known examples of value semigroups disproving the conjecture, since for all the known counterexamples it seems impossible to find suitable rings having them as value semigroups. This fact may suggest that the Wilf conjecture is more related to the structure inherited from the rings than on the combinatorical properties of these objects.