Almost canonical ideals and GAS numerical semigroups
aa r X i v : . [ m a t h . A C ] M a r ALMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS
MARCO D’ANNA AND FRANCESCO STRAZZANTI
Abstract.
We propose the notion of GAS numerical semigroup which generalizes both almost sym-metric and 2-AGL numerical semigroups. Moreover, we introduce the concept of almost canonicalideal which generalizes the notion of canonical ideal in the same way almost symmetric numeri-cal semigroups generalize symmetric ones. We prove that a numerical semigroup with maximalideal M and multiplicity e is GAS if and only if M − e is an almost canonical ideal of M − M .This generalizes a result of Barucci about almost symmetric semigroups and a theorem of Chau,Goto, Kumashiro, and Matsuoka about 2-AGL semigroups. We also study the transfer of the GASproperty from a numerical semigroup to its gluing, numerical duplication and dilatation. Introduction
The notion of Gorenstein ring turned out to have great importance in commutative algebra,algebraic geometry and other mathematics areas and in the last decades many researchers havedeveloped generalizations of this concept obtaining rings with similar properties in certain respects.With this aim, in 1997 Barucci and Fr¨oberg [3] introduced the notion of almost Gorenstein ring,inspired by numerical semigroup theory. We recall that a numerical semigroup S is simply anadditive submonoid of the set of the natural numbers N with finite complement in N . The simplestway to relate it to ring theory is by associating with S the ring k [[ S ]] = k [[ t s | s ∈ S ]], where k isa field and t is an indeterminate. Actually it is possible to associate a numerical semigroup v ( R )with every one-dimensional analytically irreducible ring R . In this case a celebrated result of Kunz[17] ensures that R is Gorenstein if and only if v ( R ) is a symmetric semigroup, see also [5, Theorem4.4.8] for a proof in the particular case of k [[ S ]]. In [3] the notions of almost symmetric numericalsemigroup and almost Gorenstein ring are introduced, where the latter is limited to analyticallyunramified rings. It turns out that k [[ S ]] is almost Gorenstein if and only if S is almost symmetric.More recently this notion has been generalized in the case of one-dimensional local ring [13] andin higher dimension [14]. Moreover, in [6] it is introduced the notion of n -AGL ring in order tostratify the Cohen-Macaulay rings. Indeed a ring is almost Gorenstein if and only if it is either1-AGL or 0-AGL, with 0-AGL equivalent to be Gorenstein. In this respect 2-AGL rings are nearto be almost Gorenstein and for this reason their properties have been deepened in [6, 11]. In [6]it is also studied the numerical semigroup case, where 2-AGL numerical semigroups are close to bealmost symmetric. Date : March 31, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Almost symmetric numerical semigroup, almost Gorenstein ring, 2-AGL semigroup, 2-AGL ring, canonical ideal.The authors were supported by the project “Propriet`a algebriche locali e globali di anelli associati a curve eipersuperfici” PTR 2016-18 - Dipartimento di Matematica e Informatica - Universit`a degli Studi di CataniaThe second author was also supported by INdAM, more precisely he was “titolare di un Assegno di Ricerca dell’IstitutoNazionale di Alta Matematica”.
In this paper we introduce the class of
Generalized Almost Symmetric numerical semigroups ,briefly GAS numerical semigroups, that includes symmetric, almost symmetric and 2-AGL numer-ical semigroups, but not 3-AGL. Moreover, if S has maximal embedding dimension and it is GAS,then it is either almost symmetric or 2-AGL. Our original motivation to introduce this class is aresult on 2-AGL numerical semigroups that partially generalize a property of almost symmetricsemigroups. More precisely, let S be a numerical semigroup with multiplicity e and let M beits maximal ideal. In [3, Corollary 8] it is proved that M − M is symmetric if and only if S isalmost symmetric with maximal embedding dimension. If we do not assume that S has maximalembedding dimension, it holds that S is almost symmetric if and only if M − e is a canonical idealof M − M (indeed S has maximal embedding dimension exactly when M − e = M − M , see [1,Theorem 5.2]). In [6, Corollary 5.4] it is shown that S is 2-AGL if and only if M − M is almostsymmetric and not symmetric, provided that S has maximal embedding dimension.Hence, it is natural to investigate what happens to M − M , for a 2-AGL semigroup, if we donot make any assumptions on its embedding dimension. It turns out that M − e is an ideal of M − M that satisfies some equivalent conditions, that are the analogue for ideals to the definingconditions of almost symmetric semigroup (cf. Definition 2.1 and Proposition 2.4); for this reasonwe called the ideals in this class almost canonical ideals . However the converse is not true: thereexist numerical semigroups S such that M − e is an almost canonical ideal of M − M , but that arenot 2-AGL. This fact lead us to look for those numerical semigroup satisfying this property, andwe found that these semigroups naturally generalize 2-AGL semigroups (this is evident if we lookat 2 K \ K , where K is the canonical ideal of S , cf. Proposition 3.1 and Definition 3.2); moreover,as we said above this class coincides with the union of 2-AGL and almost symmetric semigroups,if we assume maximal embedding dimension; hence we called them Generalized Almost Symmetric(briefly GAS). It turns out that GAS semigroups are interesting under many aspects; for example,if S is GAS, it is possible to control both the semigroup generated by its canonical ideal (that playsa fundamental role in [6]; cf. Theorem 3.7) and its pseudo-Frobenius numbers (cf. Proposition 3.8).Hence, in this paper, after recalling the basic definitions and notations, we introduce, in Section2, the concept of almost canonical ideal. We show under which respect they are a generalizationof canonical ideals and we notice that, similarly to the canonical case, a numerical semigroup S is almost symmetric if and only if it is an almost canonical ideal of itself. Moreover, we proveseveral equivalent conditions for a semigroup ideal to be almost canonical (cf. Proposition 2.4) andwe show how to find all the almost canonical ideals of a numerical semigroup and to count them(Corollary 2.6).In Section 3 we develop the theory of GAS semigroups proving many equivalent conditions (seeProposition 3.5), exploring their properties (cf. Theorem 3.7 and Proposition 3.8) and relatingthem with other classes of numerical semigroups that have been recently introduced to generalizealmost symmetric semigroups. The main result is Theorem 3.13, where it is proved that S is GASif and only if M − e is an almost canonical ideal of M − M .Finally in Section 4 we study the transfer of the GAS property from S to some numerical semi-group constructions: gluing in Theorem 4.1, numerical duplication in Theorem 4.7 and dilatationin Proposition 4.9.Several computations are performed by using the GAP system [9] and, in particular, the Numer-icalSgps package [8]. 1. Notation and basic definitions
A numerical semigroup S is a submonoid of the natural numbers N such that | N \ S | < ∞ .Therefore, there exists the maximum of N \ S that is said to be the Frobenius number of S and it LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 3 is denoted by F( S ). Given s , . . . , s ν ∈ N we set h s , . . . , s ν i = { λ s + · · · + λ ν s ν | λ , . . . , λ ν ∈ N } which is a numerical semigroup if and only if gcd( s , . . . , s ν ) = 1. We say that s , . . . , s ν are minimalgenerators of h s , . . . , s ν i if it is not possible to delete one of them obtaining the same semigroup.It is well-known that a numerical semigroup have a unique system of minimal generators, which isfinite, and its cardinality is called embedding dimension of S . The minimum non-zero element of S is said to be the multiplicity of S and we denote it by e . It is always greater than or equal to theembedding dimension of S and we say that S has maximal embedding dimension if they are equal.Unless otherwise specified, we assume that S = N .A set I ⊆ Z is said to be a relative ideal of S if I + S ⊆ I and there exists z ∈ S such that z + I ⊆ S . If it is possible to chose z = 0, i.e. I ⊆ S , we simply say that I is an ideal of S . Twovery important relative ideals are M ( S ) = S \ { } , which is an ideal and it is called the maximalideal of S , and K ( S ) = { x ∈ N | F( S ) − x / ∈ S } . We refer to the latter as the standard canonicalideal of S and we say that a relative ideal I of S is canonical if I = x + K ( S ) for some x ∈ Z . Ifthe semigroup is clear from the context, we write M and K in place of M ( S ) and K ( S ). Giventwo relative ideals I and J of S , we set I − J = { x ∈ Z | x + J ⊆ I } which is a relative ideal of S .For every relative ideal I it holds that K − ( K − I ) = I , in particular K − ( K − S ) = S . Moreover,an element x is in I if and only if F( S ) − x / ∈ K − I , see [16, Hilfssatz 5]. As a consequence weget that the cardinalities of I and K − I are equal. Also, if I ⊆ J are two relative ideals, then | J \ I | = | ( K − I ) \ ( K − J ) | . We now collect some important definitions that we are going togeneralize in the next section. Definition 1.1.
Let S be a numerical semigroup.(1) The pseudo-Frobenius numbers of S are the elements of the set PF( S ) = ( S − M ) \ S .(2) The type of S is t ( S ) = | PF( S ) | .(3) S is symmetric if and only if S = K .(4) S is almost symmetric if and only if S − M = K ∪ { F( S ) } .We note that M − M = S ∪ PF( S ). Given 0 ≤ i ≤ e −
1, let ω i be the smallest element of S that iscongruent to i modulo e . A fundamental tool in numerical semigroup theory is the so-called Ap´eryset of S that is defined as Ap( S ) = { ω = 0 , ω , . . . , ω e − } . In Ap( S ) we define the partial ordering x ≤ S y if and only if y = x + s for some s ∈ S and we denote the maximal elements of Ap( S )with respect to ≤ S by Max ≤ S (Ap( S )). With this notation PF( S ) = { ω − e | ω ∈ Max ≤ S (Ap( S )) } ,see [21, Proposition 2.20]. We also recall that S is symmetric if and only if t ( S ) = 1, that is alsoequivalent to say that k [[ S ]] has type 1 for every field k , i.e. k [[ S ]] is Gorenstein. Also for almostsymmetric semigroups many useful characterizations are known, for instance it is easy to see thatour definition is equivalent to M + K ⊆ M , but see also [19, Theorem 2.4] for another usefulcharacterization related to the Ap´ery set of S and its pseudo-Frobenius numbers.2. Almost canonical ideals of a numerical semigroup If I is a relative ideal of S , the set Z \ I has a maximum that we denote by F( I ). We set e I = I + (F( S ) − F( I )), that is the unique relative ideal J isomorphic to I for which F( S ) = F( J ),and we note that e I ⊆ K ⊆ N for every I . The following is a generalization of Definition 1.1. Definition 2.1.
Let I be a relative ideal of a numerical semigroup S .(1) The pseudo-Frobenius numbers of I are the elements of the set PF( I ) = ( I − M ) \ I .(2) The type of I is t ( I ) = | PF( I ) | .(3) I is canonical if and only if e I = K .(4) I is almost canonical if and only if e I − M = K ∪ { F( S ) } . MARCO D’ANNA AND FRANCESCO STRAZZANTI
Remark 2.2. 1. S is an almost canonical ideal of itself if and only if it is an almost symmetricsemigroup. M is an almost canonical ideal of S if and only if S is an almost symmetric semigroup. Indeed, M − M = S − M , since S = N . Moreover, t ( M ) = t ( S ) + 1. It holds that K − M = K ∪{ F( S ) } . One containment is trivial, so let x ∈ (( K − M ) \ ( K ∪{ F( S ) } )).Then 0 = F( S ) − x ∈ S and, thus, F( S ) = (F( S ) − x )+ x ∈ M +( K − M ) ⊆ K yields a contradiction.In particular, a canonical ideal is almost canonical. Since F( S ) = F( e I ), it is always in e I − M . Moreover, we claim that ( e I − M ) ⊆ K ∪{ F( S ) } . Indeed,if x ∈ ( e I − M ) \ { F( S ) } and x / ∈ K , then F( S ) − x ∈ M and F( e I ) = F( S ) = (F( S ) − x ) + x ∈ e I . Inaddition, e I is always contained in e I − M because it is a relative ideal of S . Hence, I is an almostcanonical ideal of S if and only if K \ e I ⊆ ( e I − M ).Given a relative ideal I of S , the Ap´ery set of I is Ap( I ) = { i ∈ I | i − e / ∈ I } . As in thesemigroup case, in Ap( I ) we define the partial ordering x ≤ S y if and only if y = x + s for some s ∈ S and we denote by Max ≤ S (Ap( I )) the maximal elements of Ap( I ) with respect to ≤ S . Proposition 2.3.
Let I be a relative ideal of S . The following statements hold: (1) PF( I ) = { i − e | i ∈ Max ≤ S (Ap( I )) } ; (2) I is canonical if and only if its type is .Proof. (1) An integer i ∈ I is in Max ≤ S (Ap( I )) if and only if i − e / ∈ I and s + i / ∈ Ap( I ), i.e. s + i − e ∈ I , for every s ∈ M . This is equivalent to say that i − e ∈ ( I − M ) \ I = PF( I ).(2) Since F( S ) ∈ e I − M , we have t ( e I ) = t ( I ) = 1 if and only if e I − M = e I ∪ { F( S ) } . Therefore, acanonical ideal has type 1 by Remark 2.2.3. Conversely, assume that t ( e I ) = 1 and let x / ∈ e I . Since e I ⊆ K , we only need to prove that x / ∈ K . By (1), there is a unique maximal element in Ap( e I )with respect to ≤ S and, clearly, it is F( S ) + e . Let 0 = λ ∈ N be such that x + λe ∈ Ap( e I ). Then,there exists y ∈ S such that x + λe + y = F( S ) + e and x = F( S ) − ( y + ( λ − e ) / ∈ K , since y + ( λ − e ∈ S . (cid:3) Let g ( S ) = | N \ S | denote the genus of S and let g ( I ) = | N \ e I | . We recall that 2 g ( S ) ≥ F( S )+ t ( S )and the equality holds if and only if S is almost symmetric, see, e.g., [19, Proposition 2.2 andProposition-Definition 2.3]. Proposition 2.4.
Let I be a relative ideal of S . Then g ( I ) + g ( S ) ≥ F( S ) + t ( I ) . Moreover, thefollowing conditions are equivalent: (1) I is almost canonical; (2) g ( I ) + g ( S ) = F( S ) + t ( I ) ; (3) e I − M = K − M ; (4) K − ( M − M ) ⊆ e I ; (5) If x ∈ PF( I ) \ { F( I ) } , then F( I ) − x ∈ PF( S ) .Proof. Clearly, t ( I ) = t ( e I ) and g ( I ) − t ( e I ) = | N \ e I | − | ( e I − M ) \ e I | = | N \ ( e I − M ) | . Moreover,since F( S ) + 1 − g ( S ) is the number of the elements of S smaller than F( S ) + 1, it holds thatF( S ) − g ( S ) = | N \ K | − | N \ ( K ∪ F( S )) | . We have e I − M ⊆ K ∪ { F( S ) } by Remark 2.2.4,then g ( I ) − t ( I ) ≥ F( S ) − g ( S ) and the equality holds if and only if e I − M = K ∪ { F( S ) } , i.e. I isalmost canonical. Hence, (1) ⇔ (2).(1) ⇔ (3). We have already proved that K − M = K ∪ { F( S ) } in Remark 2.2.3.(1) ⇒ (4). The thesis is equivalent to M − M ⊇ K − e I . Let x ∈ K − e I and assume by contradictionthat there exists m ∈ M such that x + m / ∈ M . Then, F( S ) − x − m ∈ K ∪ { F( S ) } = e I − M and, LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 5 so, F( S ) − x ∈ e I . Since x ∈ K − e I , this implies F( S ) ∈ K , that is a contradiction.(4) ⇒ (1). Let x ∈ K . It is enough to prove that x ∈ e I − M . Suppose by contradiction that thereexists m ∈ M such that x + m / ∈ e I ⊇ K − ( M − M ). In particular, x + m / ∈ K − ( M − M ) and soF( S ) − ( x + m ) ∈ M − M . This implies F( S ) − x ∈ M , that is a contradiction because x ∈ K .(1) ⇒ (5) We notice that PF( e I ) = { x + F( S ) − F( I ) | x ∈ PF( I ) } . Let x ∈ PF( I ) \ { F( I ) } and let y = x + F( S ) − F( I ) ∈ PF( e I ) \ { F( S ) } . We first note that F( S ) − y / ∈ S , otherwiseF( S ) = y + (F( S ) − y ) ∈ e I . Assume by contradiction that F( S ) − y / ∈ PF( S ), i.e. there exists m ∈ M such that F( S ) − y + m / ∈ S . This implies that y − m ∈ K ⊆ e I − M by (1) and, thus, y = ( y − m ) + m ∈ e I yields a contradiction. Hence, F( I ) − x = F( S ) − y ∈ PF( S ).(5) ⇒ (4) Assume by contradiction that there exists x ∈ ( K − ( M − M )) \ e I . It easily follows from thedefinition that there is s ∈ S such that x + s ∈ PF( e I ). Then, F( S ) − x − s ∈ PF( S ) ∪ { } ⊆ M − M by (5) and F( S ) − s = x +(F( S ) − x − s ) ∈ ( K − ( M − M ))+( M − M ) ⊆ K gives a contradiction. (cid:3) Remark 2.5. 1.
In [19, Theorem 2.4] it is proved that a numerical semigroup S is almost symmetricif and only if F( S ) − f ∈ PF( S ) for every f ∈ PF( S ) \ { F( S ) } . Hence, the last condition ofProposition 2.4 can be considered a generalization of this result. Almost canonical ideals naturally arise characterizing the almost symmetry of the numericalduplication S ✶ b I of S with respect to the ideal I and b ∈ S , a construction introduced in [7].Indeed [7, Theorem 4.3] says that S ✶ b I is almost symmetric if and only if I is almost canonicaland K − e I is a numerical semigroup. Let T be an almost symmetric numerical semigroup with odd Frobenius number (or, equivalently,odd type). Let b be an odd integer such that 2 b ∈ T and set I = { x ∈ Z | x + b ∈ T } . Then,[22, Proposition 3.3] says that T can be realized as a numerical duplication T = S ✶ b I , where S = T / { y ∈ Z | y ∈ T } , while [22, Theorem 3.7] implies that I is an almost canonical ideal of S . In general this is not true if the Frobenius number of T is even.Since F( K − ( M − M )) = F( e I ) = F( K ) and e I ⊆ K for every relative ideal I , Condition (4) ofProposition 2.4 allows to find all the almost canonical ideals of a numerical semigroup. Clearly itis enough to focus on the relative ideals with Frobenius number F( S ). Corollary 2.6.
Let S be a numerical semigroup with type t . If I is almost canonical, then t ( I ) ≤ t + 1 . Moreover, for every integer i such that ≤ i ≤ t + 1 , there are exactly (cid:0) ti − (cid:1) almost canonicalideals of S with Frobenius number F( S ) and type i . In particular, there are exactly t almostcanonical ideals of S with Frobenius number F( S ) .Proof. Let C = { s ∈ S | s > F( S ) } = K − N be the conductor of S and let n ( S ) = |{ s ∈ S | s < F( S ) }| . It is straightforward to see that g ( S ) + n ( S ) = F( S ) + 1. If I is almost canonical,Proposition 2.4 implies that t ( I ) = g ( I ) + g ( S ) − F( S ) ≤ | N \ ( K − ( M − M )) | − n ( S ) + 1 == | ( M − M ) \ ( K − N ) | − n ( S ) + 1 = | ( M − M ) \ C | − n ( S ) + 1 == | ( M − M ) \ S | + | S \ C | − n ( S ) + 1 = t + n ( S ) − n ( S ) + 1 = t + 1 . By Proposition 2.4 an ideal I with F( I ) = F( S ) is almost canonical if and only if K − ( M − M ) ⊆ I ⊆ K and we notice that | K \ ( K − ( M − M )) | = | ( M − M ) \ S | = t . Let A ⊆ ( K \ ( K − ( M − M )))and consider I = ( K − ( M − M )) ∪ A . We claim that I is an ideal of S . Indeed, let x ∈ A , m ∈ M and y ∈ ( M − M ). It follows that m + y ∈ M and, then, x + m + y ∈ K , since K is anideal. Therefore, x + m ∈ K − ( M − M ) and I is an ideal of S . Moreover, by [7, Lemma 4.7], MARCO D’ANNA AND FRANCESCO STRAZZANTI t ( I ) = | ( K − I ) \ S | + 1 = | K \ I | + 1 = t + 1 − | A | and the thesis follows, because there are (cid:0) ti − (cid:1) subsets of K \ ( K − ( M − M )) with cardinality t + 1 − i . (cid:3) If S is a symmetric semigroup, the only almost canonical ideals with Frobenius number F( S ) are M and S . In this case t ( M ) = t ( S ) + 1 = 2. If S is pseudo-symmetric, the four almost canonicalideals with Frobenius number F( S ) are M , S , M ∪ { F( S ) / } and K . In this case t ( M ) = 3, t ( S ) = t ( M ∪ { F( S ) / } ) = 2 and t ( K ) = 1.3. GAS numerical semigroups
In [6] it is introduced the notion of n -almost Gorenstein local rings, briefly n -AGL rings, where n is a non-negative integer. These rings generalize almost Gorenstein ones that are obtained wheneither n = 0, in which case the ring is Gorenstein, or n = 1. In particular, in [6] it is studied thecase of the 2-AGL rings, that are closer to be almost Gorenstein, see also [11].Given a numerical semigroup S with standard canonical ideal K we denote by h K i the numericalsemigroup generated by K . Following [6] we say that S is n -AGL if |h K i \ K | = n . It follows that S is symmetric if and only if it is 0-AGL, whereas it is almost symmetric and not symmetric if andonly if it is 1-AGL.It is easy to see that a numerical semigroup is 2-AGL if and only if 2 K = 3 K and | K \ K | = 2, see[6, Theorem 1.4] for a proof in a more general context. We now give another easy characterizationthat will lead us to generalize this class. Proposition 3.1.
A numerical semigroup S is 2-AGL if and only if K = 3 K and K \ K = { F( S ) − x, F( S ) } for a minimal generator x of S .Proof. One implication is trivial, so assume that S is 2-AGL. Since S is not symmetric, there exists k ∈ N such that k and F( S ) − k are in K and so F( S ) ∈ K \ K . Let now a ∈ (2 K \ K ) \ { F( S ) } .Since a / ∈ K , we have F( S ) − a ∈ S . Assume that F( S ) − a = s + s with s , s ∈ S \ { } . Itfollows that F( S ) − s = a + s ∈ K , since 2 K is a relative ideal, and by definition F( S ) − s / ∈ K .Therefore, { a, F( S ) − s , F( S ) } ⊆ K \ K and this is a contradiction, since S is 2-AGL. Hence, a = F( S ) − x , where x is a minimal generator of S . (cid:3) In light of the previous proposition we propose the following definition.
Definition 3.2.
We say that S is a generalized almost symmetric numerical semigroup, brieflyGAS numerical semigroup, if either 2 K = K or 2 K \ K = { F( S ) − x , . . . , F( S ) − x r , F( S ) } forsome r ≥ x , . . . , x r of S such that x i − x j / ∈ PF( S ) for every i, j .The last condition could seem less natural, but these semigroups have a better behaviour. Forinstance, in Theorem 3.7 we will see that this condition ensures that every element in h K i \ K canbe written as F( S ) − x for a minimal generator x of S .We recall that S is symmetric if and only if 2 K = K and it is almost symmetric exactly when2 K \ K ⊆ { F( S ) } . Examples 3.3. 1.
Let S = h , , , , i . Then, PF( S ) = { , , , } and 2 K \ K = { − , − , − , − , − , } . Hence, S is a GAS semigroup. If S = h , , i , we have 2 K = 3 K and 2 K \ K = { − , − , } . Hence, S is 3-AGL butit is not GAS because 14 is not a minimal generator of S . Consider the semigroup S = h , , , , , , , i . We have 2 K \ K = { − , − , } ,but S is not GAS because 11 − ∈ PF( S ). In this case 2 K = 3 K and thus S is 3-AGL. LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 7
The last example shows that in a numerical semigroup S with maximal embedding dimensionthere could be many minimal generators x such that F( S ) − x ∈ K \ K . This is not the case if weassume that S is GAS. Proposition 3.4. If S has maximal embedding dimension e and it is GAS , then it is either almostsymmetric or with K \ K = { F( S ) − e, F( S ) } .Proof. Assume that S is not almost symmetric and let F( S ) − x = k + k ∈ K \ K with x = 0 and k , k ∈ K . Let x = e and consider F( S ) − e = k + k + x − e . Since x − e ≤ F( S ) − e < F( S ) and S has maximal embedding dimension, x − e ∈ PF( S ) \ { F( S ) } ⊆ K and, therefore, F( S ) − e ∈ K \ K .Moreover, F( S ) − e cannot be in 2 K , because S is GAS and x − e ∈ PF( S ), then, k + x − e ∈ K \ K .Hence, we have F( S ) − (F( S ) − k − x + e ) ∈ K \ K and, thus, F( S ) − k − x + e is a minimalgenerator of S . Since S has maximal embedding dimension, this implies that F( S ) − k − x ∈ PF( S )and, then, F( S ) − k ∈ S yields a contradiction, since k ∈ K . This means that x = e and2 K \ K = { F( S ) − e, F( S ) } .Suppose by contradiction that 2 K = 3 K and let F( S ) − y ∈ K \ K . In particular, F( S ) − y / ∈ K and, therefore, y ∈ S . If F( S ) − y = k + k + k with k i ∈ K for every i , then k + k ∈ K \ K and, thus, k + k = F( S ) − e . This implies that F( S ) − e < F( S ) − y , i.e. y < e , that is acontradiction. (cid:3) In particular, we note that in a 2-AGL semigroup with maximal embedding dimension it alwaysholds that 2 K \ K = { F( S ) − e, F( S ) } . Proposition 3.5.
Given a numerical semigroup S , the following conditions are equivalent: (1) S is GAS ; (2) x − y / ∈ ( M − M ) for every different x, y ∈ M \ ( S − K ) ; (3) either S is symmetric or M ⊆ S − K ⊆ M and M − M = (( S − K ) − M ) ∪ { } .Proof. If S is symmetric, then M ⊆ S − K and both (1) and (2) are true, so we assume S = K .(1) ⇒ (2) Note that K − S = K and K − ( S − K ) = K − (( K − K ) − K ) = K − ( K − K ) = 2 K .Thus, x ∈ S \ ( S − K ) if and only if F( S ) − x ∈ ( K − ( S − K )) \ ( K − S ) = 2 K \ K . Hence, if S isGAS, then x − y / ∈ S ∪ PF( S ) = M − M for every x, y ∈ M \ ( S − K ).(2) ⇒ (1) If x , y ∈ M \ ( S − K ), then F( S ) − x , F( S ) − y ∈ K \ K and x − y / ∈ PF( S ), since itis not in M − M . We only need to show that x is a minimal generator of S . If by contradiction x = s + s , with s , s ∈ M , it follows that also s is in M \ ( S − K ). Therefore, s = x − s ∈ M yields a contradiction since x − s / ∈ M − M by hypothesis.(2) ⇒ (3) Since S is not symmetric, S − K is contained in M . Moreover, if 2 M is not in S − K ,then there exist m , m ∈ M such that m + m ∈ M \ ( S − K ). Clearly also m is not in S − K and ( m + m ) − m = m ∈ M ⊆ M − M yields a contradiction.It always holds that (( S − K ) − M ) ∪ { } ⊆ M − M , then given x ∈ ( M − M ) \ { } and m ∈ M , we only need to prove that x + m ∈ S − K . If m ∈ M \ ( S − K ) and x + m / ∈ S − K ,then ( x + m ) − m = x ∈ M − M gives a contradiction. If m ∈ ( S − K ) \ M and k ∈ K , then0 = m + k ∈ S and, so, x + m + k ∈ M , that implies x + m ∈ S − K . Finally, if m ∈ M , then x + m ∈ M ⊆ S − K .(3) ⇒ (2) Let x, y ∈ M \ ( S − K ) with x = y and assume by contradiction that x − y ∈ ( M − M ) =(( S − K ) − M ) ∪ { } . By hypothesis y ∈ M , then x = ( x − y ) + y ∈ S − K yields a contradiction. (cid:3) In the definition of GAS semigroup we required that in 2 K \ K there are only elements of thetype F( S ) − x with x minimal generator of S . In general, this does not imply that the elementsin 3 K \ K are of the same type. For instance, consider S = h , , , , , , , i , where MARCO D’ANNA AND FRANCESCO STRAZZANTI K \ K = { − , − , − , − , } and 3 K \ K = { − , − } . However, byProposition 3.4, this semigroup is not GAS. In fact, this never happens in a GAS semigroup as weare going to show in Theorem 3.7. First we need a lemma. Lemma 3.6.
Assume that K \ K = { F( S ) − x , . . . , F( S ) − x r , F( S ) } with x , . . . , x r minimalgenerators of S . If F( S ) − x ∈ nK \ ( n − K for some n > and x = s + s with s , s ∈ M ,then F( S ) − s ∈ ( n − K .Proof. Let F( S ) − ( s + s ) = k + · · · + k n ∈ nK \ ( n − K with k i ∈ K for 1 ≤ i ≤ n . SinceF( S ) − ( s + s ) / ∈ ( n − K , we have F( S ) = k + k ∈ K \ K and, then, F( S ) − ( k + k )is a minimal generator of S . Since F( S ) − ( k + k ) = s + s + k + · · · + k n , this implies that s + k + · · · + k n / ∈ S , that is k + k + s = F( S ) − ( s + k + · · · + k n ) ∈ K . Therefore,F( S ) − s = ( k + k + s ) + k + · · · + k n ∈ ( n − K and the thesis follows. (cid:3) Theorem 3.7.
Let S be a GAS numerical semigroup that is not symmetric. Then, h K i \ K = { F( S ) − x , . . . , F( S ) − x r , F( S ) } for some minimal generators x , . . . , x r with r ≥ and x i − x j / ∈ PF( S ) for every i and j .Proof. We first prove that x i − x j / ∈ PF( S ) for every i and j without assuming that x i and x j areminimal generators. We can suppose that x i = x and x j = x .Let F( S ) − x = k + · · · + k n ∈ nK \ ( n − K with k i ∈ K for every i and assume by contradictionthat x − x ∈ PF( S ). We note that F( S ) − x = k + · · · + k n + ( x − x ) and k + ( x − x ) ∈ K .Indeed, if F( S ) − k − ( x − x ) = s ∈ S , then s = 0 and F( S ) − k = ( x − x ) + s ∈ S yields acontradiction. If k + k +( x − x ) / ∈ K , then it is in 2 K \ K and, since also k + k ∈ K \ K , we get acontradiction because their difference is a pseudo-Frobenius number. Hence, k + k +( x − x ) ∈ K .We proceed by induction on n . If n = 2, it follows that F( S ) − x = k + k + ( x − x ) ∈ K , thatis a contradiction. So, let n ≥ i be the minimum index for which k + · · · + k i + ( x − x ) / ∈ K . It follows that k + · · · + k i + ( x − x ) ∈ K \ K and, since also k + k ∈ K \ K , thisimplies that k + · · · + k i + ( x − x ) / ∈ PF( S ). Moreover, it cannot be in S , because it is thedifference of two minimal generators, since S is GAS. Therefore, there exists m ∈ M such that k + · · · + k i + ( x − x ) + m / ∈ S , that means F( S ) − ( k + · · · + k i + ( x − x ) + m ) = k ′ ∈ K .Thus, F( S ) − (( x − x ) + m ) = k ′ + k + · · · + k i ∈ jK \ K for some 1 < j < n . Moreover,F( S ) − m = k ′ + k + · · · + k i + ( x − x ) ∈ h K i \ K and by induction ( x − x ) + m − m / ∈ PF( S ),that is a contradiction. Hence, x − x / ∈ PF( S ).Let now h ≥
3. To prove the theorem it is enough to show that, if F( S ) − x ∈ hK \ ( h − K ,then x is a minimal generators of S . We proceed by induction on h . Using the GAS hypothesis,the case h = 3 is very similar to the general case, so we omit it (the difference is that alsoF( S ) ∈ K \ K ). Suppose by contradiction that x = s + s and F( S ) − ( s + s ) = k + · · · + k h ∈ hK \ ( h − K with k , . . . , k h ∈ K and s , s ∈ M . Clearly, F( S ) − s / ∈ K and by Lemma 3.6we have F( S ) − s ∈ ( h − K ; in particular, s is a minimal generator of S by induction. Let1 < i < h be such that F( S ) − s ∈ iK \ ( i − K . Since F( S ) − ( s + s ) / ∈ ( h − K , we have k + · · · + k i ∈ iK \ ( i − K and, by induction, F( S ) − ( k + · · · + k i ) is a minimal generator of S and F( S ) − ( k + · · · + k i ) − s / ∈ PF( S ) by the first part of the proof. This means that there exists s ∈ M such that F( S ) − ( k + · · · + k i ) − s + s / ∈ S , i.e. k + · · · + k i + s − s ∈ K . This implies thatF( S ) − ( s + s ) = ( k + · · · + k i + s − s ) + k i +1 + · · · + k h ∈ ( h − i + 1) K and, since h − i + 1 < h ,the induction hypothesis yields a contradiction because s + s is not a minimal generator of S . (cid:3) We recall that in an almost symmetric numerical semigroup F( S ) − f ∈ PF( S ) for every f ∈ PF( S ) \ { F( S ) } , see [19, Theorem 2.4]. The following proposition generalizes this fact. LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 9
Proposition 3.8.
Let S be a numerical semigroup with K \ K = { F( S ) − x , . . . , F( S ) − x r , F( S ) } ,where x i is a minimal generator of S for every i . (1) For every i , there exist f j , f k ∈ PF( S ) such that f j + f k = F( S ) + x i . (2) For every f ∈ PF( S ) \ { F( S ) } , it holds either F( S ) − f ∈ PF( S ) or F( S ) − f + x i ∈ PF( S ) for some i .Proof. Let F( S ) − x i = k + k ∈ K \ K for some k , k ∈ K and let s ∈ M . Since x i + s ∈ S , we haveF( S ) − x i − s / ∈ K and then F( S ) − x i − s = k + k − s / ∈ K because x i + s is not a generator of S . Inparticular, k − s and k − s are not in K . This means that F( S ) − k + s and F( S ) − k + s are in S and,thus, F( S ) − k , F( S ) − k ∈ PF( S ). Moreover, F( S ) − k +F( S ) − k = 2 F( S ) − (F( S ) − x i ) = F( S )+ x i and (1) holds.Let now f ∈ PF( S ) \ { F( S ) } and assume that F( S ) − f / ∈ PF( S ). Then, there exists s ∈ M suchthat F( S ) − f + s ∈ PF( S ). In particular, f − s ∈ K and F( S ) − s = (F( S ) − f ) + ( f − s ) ∈ K \ K ;thus, s has to be equal to x i for some i and F( S ) − f + x i ∈ PF( S ). (cid:3) Examples 3.9. 1.
Let S = h , , , , i . We have 2 K \ K = { − , } and S is 2-AGL.In this case PF( S ) = { , , , , } and 100 + 209 = 132 + 177 = 281 + 28. Consider S = h , , , , i . Here 2 K \ K = { − , } and the semigroup is 2-AGL. Moreover, PF( S ) = { , , , , , , } , 218 + 267 = 226 + 259 = 485 and249 + 322 = 485 + 86. If S = h , , , i , then 2 K \ K = { − , − , − , − , } and PF( S ) = { , , , , } . Hence, S is GAS and, according to the previous proposition, we haveF( S ) + 9 = 11 + 15 F( S ) + 12 = 14 + 15F( S ) + 10 = 11 + 16 F( S ) + 13 = 14 + 16 . Conditions (1) and (2) in Proposition 3.8 do not imply that every x i is a minimal generator. Forinstance, if we consider the numerical semigroup S = { , , , , } , we have 2 K \ K = { − , − , − , − , − , − , − , − , } and PF( S ) = { , , , , , } .Moreover, F( S ) + 40 = 41 + 41 F( S ) + 20 = 29 + 33F( S ) + 36 = 37 + 41 F( S ) + 19 = 28 + 33F( S ) + 32 = 37 + 37 F( S ) + 16 = 29 + 29F( S ) + 24 = 33 + 33 F( S ) + 15 = 28 + 29and, so, it is straightforward to see that the conditions in Proposition 3.8 hold, but 32, 36 and 40are not minimal generators.We recall that L( S ) denotes the set of the gaps of the second type of S , i.e. the integers x suchthat x / ∈ S and F( S ) − x / ∈ S , i.e. x ∈ K \ S , and that S is almost symmetric if and only ifL( S ) ⊆ PF( S ), see [3]. Lemma 3.10.
Let S be a numerical semigroup with K \ K = { F( S ) − x , . . . , F( S ) − x r , F( S ) } ,where x i is a minimal generator of S for every i . If x ∈ L( S ) and F( S ) − x / ∈ PF( S ) , then both x and F( S ) − x + x i are pseudo-Frobenius numbers of S for some i .Proof. Assume by contradiction that x / ∈ PF( S ). Therefore, there exists s ∈ M such that x + s / ∈ S and, then, F( S ) − x − s ∈ K . Moreover, since F( S ) − x / ∈ PF( S ), there exists t ∈ M such thatF( S ) − x + t / ∈ S and then x − t ∈ K . Consequently, F( S ) − s − t = (F( S ) − x − s ) + ( x − t ) ∈ K and F( S ) − s − t / ∈ K , since s + t ∈ S . This is a contradiction, because s + t is not a minimal generator of S . Hence, x ∈ PF( S ) and, since F( S ) − x / ∈ PF( S ), Proposition 3.8 implies thatF( S ) − x + x i ∈ PF( S ) for some i . (cid:3) Lemma 3.11.
As ideal of M − M , it holds ^ M − e = M − e and K ( M − M ) \ ( M − e ) = { x − e | x ∈ L( S ) and F( S ) − x / ∈ PF( S ) } . Proof.
We notice that F( S ) − e / ∈ ( M − M ) and, if y > F( S ) − e and m ∈ M , we have y + m > F( S ) − e + m ≥ F( S ). Therefore, F( M − M ) = F( S ) − e = F( M − e ) and, then, ^ M − e = M − e .We have x − e ∈ K ( M − M ) \ ( M − e ) if and only if x / ∈ M and (F( S ) − e ) − ( x − e ) / ∈ ( M − M )that is in turn equivalent to x / ∈ M and F( S ) − x / ∈ S ∪ PF( S ). Since x = 0, this means that x ∈ L( S ) and F( S ) − x / ∈ PF( S ). (cid:3) The following corollary was proved in [1, Theorem 5.2] in a different way.
Corollary 3.12. S is almost symmetric if and only if M − e is a canonical ideal of M − M .Proof. By definition M − e is a canonical ideal of M − M if and only if K ( M − M ) = ( M − e ). Inlight of the previous lemma, this means that there are no x ∈ L( S ) such that F( S ) − x / ∈ PF( S ),that is equivalent to say that L( S ) ⊆ PF( S ), i.e. S is almost symmetric. (cid:3) In [3, Corollary 8] it was first proved that S is almost symmetric with maximal embeddingdimension if and only if M − M is a symmetric semigroup. In general it holds M − M ⊆ M − e ⊆ K ( M − M ) and the first inclusion is an equality if and only if S has maximal embedding dimension,whereas the previous corollary says that the second one is an equality if and only if S is almostsymmetric. Moreover, if S has maximal embedding dimension, in [6, Corollary 5.4] it is provedthat S is 2-AGL if and only if M − M is an almost symmetric semigroup which is not symmetric.If we want to generalize this result in the same spirit of Corollary 3.12, it is not enough to considerthe 2-AGL semigroups, but we need that S is GAS. More precisely, we have the following result. Theorem 3.13.
The semigroup S is GAS if and only if M − e is an almost canonical ideal of thesemigroup M − M .Proof. In the light of Remark 2.2.4 and Lemma 3.11, M − e is an almost canonical ideal of M − M if and only if(1) K ( M − M ) \ ( M − e ) ⊆ (( M − e ) − (( M − M ) \ { } )) . Assume that S is GAS with 2 K \ K = { F( S ) − x , . . . , F( S ) − x r , F( S ) } . By Lemma 3.11 theelements of K ( M − M ) \ ( M − e ) can be written as x − e with x ∈ L( S ) and F( S ) − x / ∈ PF( S ). Inaddition, Lemma 3.10 implies that both x and F( S ) − x + x i are pseudo-Frobenius numbers of S for some i . Let 0 = z ∈ ( M − M ). We need to show that x − e + z ∈ M − e , i.e. x + z ∈ M . Assumeby contradiction x + z / ∈ M , which implies F( S ) − x − z ∈ K . Since x + z / ∈ M and x ∈ PF( S ), itfollows that z / ∈ M and, then, z ∈ PF( S ); hence, z + x i ∈ M and F( S ) − z − x i / ∈ K . We also havethat x − x i ∈ K , since F( S ) − x + x i ∈ PF( S ). Therefore,F( S ) − z − x i = (F( S ) − x − z ) + ( x − x i ) ∈ K \ K and this yields a contradiction because ( z + x i ) − x i ∈ PF( S ) and S is a GAS semigroup.Conversely, assume that the inclusion (1) holds. An element in 2 K \ K can be written as F( S ) − s for some s ∈ S , since it is not in K . Assume by contradiction that s = 0 is not a minimal generatorof S , i.e. F( S ) − s − s = k + k ∈ K \ K for some s , s ∈ M and k , k ∈ K . It follows that LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 11 F( S ) − k − s = k + s / ∈ S , otherwise F( S ) − s ∈ K . Moreover, k + s / ∈ PF( S ) ∪ S , since k + s + s = F( S ) − k / ∈ S . Hence, Lemma 3.11 and our hypothesis imply that k + s − e = F( S ) − k − s − e ∈ (( M − e ) − (( M − M ) \ { } )) . Therefore, F( S ) − k − e = ( k + s − e ) + s ∈ M − e and, thus, k / ∈ K yields a contradiction.This means that 2 K \ K = { F( S ) − x , . . . , F( S ) − x r , F( S ) } with x i minimal generator of S forevery i . Now, assume by contradiction that z = x i − x j ∈ PF( S ) for some i, j and let F( S ) − x i =F( S ) − x j − z = k + k for some k , k ∈ K . Since k + z + x j = F( S ) − k / ∈ S , it follows that k + z / ∈ S ∪ PF( S ). Moreover, F( S ) − k − z / ∈ S , otherwise F( S ) − k ∈ S . Therefore, Lemma3.11 and inclusion (1) imply that F( S ) − k − z − e ∈ (( M − e ) − (( M − M ) \ { } )) and, since z ∈ M − M , it follows that F( S ) − k ∈ M which is a contradiction because k ∈ K . (cid:3) Example 3.14.
Consider S = h , , , , i , that is a GAS numerical semigroup with 2 K \ K = { − , − , − , } . Then, M − M − M by the previoustheorem. In fact M − M = { , , , , , , →} ,K ( M − M ) = { , , , , , , , , , , , , →} ,M − { , , , , , , , , , →} , ( M − − (( M − M ) \ { } ) = K ( M − M ) ∪ { } = { , , , , →} . Remark 3.15. If S is GAS, it is possible to compute the type of M − e seen as an ideal of thesemigroup M − M . In fact by Theorem 3.13 and Proposition 2.4 it follows that t ( M − e ) = g ( M − e ) + g ( M − M ) − F( M − M ) == g ( M ) − e + g ( S ) − t ( S ) − F( S ) + e = 2 g ( S ) + 1 − t ( S ) − F( S ) . Moreover, we recall that 2 g ( S ) ≥ t ( S ) + F( S ) is always true and the equality holds exactly when S is almost symmetric. Therefore, as t ( S ) is a measure of how far S is from being symmetric, t ( M − e ) = t ( M ) (as ideal of M − M ) can be seen as a measure of how far S is from being almostsymmetric. On the other hand, we note that the type of M as an ideal of S is simply t ( S ) + 1.If S has type 2 and PF( S ) = { f, F( S ) } , in [6, Theorem 6.2] it is proved that S is 2-AGL ifand only if 3(F( S ) − f ) ∈ S and F( S ) = 2 f − x for some minimal generator x of S . In the nextproposition we generalize this result to the GAS case. Proposition 3.16.
Assume that S is not almost symmetric and that it has type 2, i.e. PF( S ) = { f, F( S ) } . Then, S is GAS if and only if F( S ) = 2 f − x for some minimal generator x of S . Inthis case, if n is the minimum integer for which n (F( S ) − f ) ∈ S , then | K \ K | = 2 , | K \ K | = · · · = | ( n − K \ ( n − K | = 1 and nK = ( n − K .Proof. Assume first that S is GAS and let F( S ) − x , F( S ) − y ∈ K \ K . Proposition 3.8 impliesthat F( S ) + x = f + f and F( S ) + y = f + f for some f , f , f , f ∈ PF( S ). Since f i has tobe different from F( S ) for all i , it follows that F( S ) + x = F( S ) + y = 2 f and, then, x = y . Inparticular, F( S ) = 2 f − x .Assume now that F( S ) = 2 f − x for some minimal generator x of S . Clearly, F( S ) − x =2(F( S ) − f ) ∈ K \ K . Let y = 0 , x be such that F( S ) − y ∈ K \ K . Since 2 K \ K is finite, we mayassume that y is maximal among such elements with respect to ≤ S , that is F( S ) − ( y + m ) / ∈ K \ K for every m ∈ M . Let F( S ) − y = k + k with k , k ∈ K . Since F( S ) − y − m = k + k − m / ∈ K \ K , then k − m and k − m are not in K , which is equivalent to F( S ) − k + m ∈ S andF( S ) − k + m ∈ S for every m ∈ M . This means that F( S ) − k , F( S ) − k ∈ PF( S ) \ { F( S ) } which implies F( S ) − y = 2(F( S ) − f ) = F( S ) − x and, thus, x = y . Therefore, | K \ K | = 2 and S is GAS.Moreover, if S is GAS and F( S ) − y = k + · · · + k r ∈ rK \ ( r − K with r > k , . . . , k r ∈ K ,then k = · · · = k r = F( S ) − f because k i + k j ∈ K \ K for every i and j . Therefore, if n (F( S ) − f ) ∈ S , then nK = ( n − K . Assume that r (F( S ) − f ) / ∈ S . Clearly, it is in rK and we claim that it is not in K . In fact, if r (F( S ) − f ) ∈ K , it follows that it is in L( S ) and, ifF( S ) − r (F( S ) − f ) = f , then ( r − S ) − f ) = 0 ∈ S yields a contradiction. Therefore, Lemma 3.10implies that F( S ) − r (F( S ) − f ) + x = f and, again, ( r − S ) − f ) = x ∈ S gives a contradiction.This means that r (F( S ) − f ) ∈ rK \ K . Moreover, if r (F( S ) − f ) = k + · · · + k r ′ ∈ r ′ K \ ( r ′ − K with 1 < r ′ < r and k , . . . , k r ′ ∈ K , we get k = · · · = k r ′ = F( S ) − f as above, that is acontradiction. Hence, | rK \ ( r − K | = 1 for every 1 < r < n . (cid:3) Example 3.17.
Consider S = h , , i . In this case f = 8 and F( S ) = 9. Therefore, the equalityF( S ) = 2 f − S is GAS. With the notation of the previous corollary we have n = 5and, in fact, 2 K \ K = { , } , 3 K \ K = { } and 4 K \ K = { } .In [15] another generalization of almost Gorenstein ring is introduced. More precisely a Cohen-Macaulay local ring admitting a canonical module ω is said to be nearly Gorenstein if the trace of ω contains the maximal ideal. In the case of numerical semigroups it follows from [15, Lemma 1.1]that S is nearly Gorenstein if and only if M ⊆ K + ( S − K ), see also the arXiv version of [15]. Itis easy to see that an almost symmetric semigroup is nearly Gorenstein, but in [6] it is noted thata 2-AGL semigroup is never nearly Gorenstein (see also [4, Remark 3.7] for an easy proof in thenumerical semigroup case). This does not happen for GAS semigroups. Corollary 3.18.
Let S be a GAS semigroup, not almost symmetric, with
PF( S ) = { f, F( S ) } . Itis nearly Gorenstein if and only if f − S ) ∈ S .Proof. We will use the following characterization proved in [18]: S is nearly Gorenstein if and onlyif for every minimal generator y of S there exists g ∈ PF( S ) such that g + y − g ′ ∈ S for every g ′ ∈ PF( S ) \ { g } .By Proposition 3.16 it follows that F( S ) = 2 f − x with x minimal generator of S . Let y = x another minimal generator of S and assume by contradiction that F( S ) + y − f / ∈ S . Therefore,there exists s ∈ S such that F( S ) + y − f + s ∈ PF( S ). If it is equal to F( S ), then f = y + s ∈ S yields a contradiction. If F( S ) + y − f + s = f , then y + s = 2 f − F( S ) = x by Proposition 3.16and this gives a contradiction, since x = y is a minimal generator of S . Hence, F( S ) + y − f ∈ S for every minimal generator y = x . On the other hand, F( S ) + x − f = 2 f − x + x − f = f / ∈ S and, therefore, S is nearly Gorenstein if and only if f + x − F( S ) = 3 f − S ) ∈ S . (cid:3) Examples 3.19. 1.
In Example 3.17 we have 3 f − S ) = 6 ∈ S and, then, the semigroup isboth GAS and nearly Gorenstein. Consider S = h , , i that has PF( S ) = { , } . Since 2 ∗ −
109 = 9 and 3 ∗ − ∗
109 = − / ∈ S , the semigroup is GAS but not nearly Gorenstein. If S = h , , , i , we have PF( S ) = { , } and 2 ∗ −
39 = 37 is not a minimal generators,thus, S is not GAS. On the other hand, it is straightforward to check that this semigroup is nearlyGorenstein. Remark 3.20.
In literature there are other two generalizations of almost Gorenstein ring. One isgiven by the so-called ring with canonical reduction, introduced in [20], which is a one-dimensionalCohen-Macaulay local ring ( R, m ) possessing a canonical ideal I that is a reduction of m . When R = k [[ S ]] is a numerical semigroup ring, this definition gives a generalization of almost symmetricsemigroup and R has a canonical reduction if and only if e + F( S ) − g ∈ S for every g ∈ N \ S , see LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 13 [20, Theorem 3.13]. This notion is unrelated with the one of GAS semigroup, in fact it is easy tosee that S = h , , , i is GAS and it doesn’t have canonical reductions, while S = h , , , i isnot GAS, but has a canonical reduction.Another generalization of the notion of almost Gorenstein ring is given by the so-called gener-alized Gorenstein ring, briefly GGL, introduced in [10, 12]. A Cohen-Macaulay local ring ( R, m )with a canonical module ω is said to be GGL with respect to a if either R is Gorenstein or thereexists an exact sequence of R -modules0 −→ R ϕ −→ ω −→ C −→ C is an Ulrich module of R with respect to some m -primary ideal a and ϕ ⊗ R/ a is injective.We note that R is almost Gorenstein and not Gorenstein if and only if it is GGL with respect to m .Let S be a numerical semigroup and order PF( S ) = { f , f , . . . , f t = F( S ) } by the usual order in N . Defining a numerical semigroup GGL if its associated ring is GGL, in [23] it is proved a usefulcharacterization: S is GGL if either it is symmetric or the following properties hold:(1) there exists x ∈ S such that f i + f t − i = F( S ) + x for every i = 1 , . . . , ⌈ t/ ⌉ ;(2) (( c − M ) ∩ S ) \ c = { x } , where c = S − h K i .Using this characterization it is not difficult to see that also this notion is unrelated with the one ofGAS semigroup. In fact, the semigroups in Examples 3.9.2 and 3.9.3 are GAS but do not satisfy (1),whereas the semigroup S = h , , i is not GAS by Proposition 3.16, because PF( S ) = { , } ,but it is easy to see that it is GGL with x = 10.4. Constructing GAS numerical semigroups
In this section we study the behaviour of the GAS property with respect to some constructions.In this way we will be able to construct many numerical semigroups satisfying this property.4.1.
Gluing of numerical semigroups.
Let S = h s , . . . , s n i and S = h t , . . . , t m i be twonumerical semigroups and assume that s , . . . , s n and t , . . . , t m are minimal generators of S and S respectively. Let also a ∈ S and b ∈ S be not minimal generators of S and S respectively andassume gcd( a, b ) = 1. The numerical semigroup h aS , bS i = h as , . . . , as n , bt , . . . , bt m i is said tobe the gluing of S and S with respect to a and b . It is well-known that as , . . . , as n , bt , . . . , bt m are its minimal generators, see [21, Lemma 9.8]. Moreover, the pseudo-Frobenius numbers of T = h aS , bS i are PF( T ) = { af + bf + ab | f ∈ PF( S ) , f ∈ PF( S ) } , see [19, Proposition 6.6]. In particular, t ( T ) = t ( S ) t ( S ) and F( T ) = a F( S ) + b F( S ) + ab .Consequently, since K ( T ) is generated by the elements F( T ) − f with f ∈ PF( T ), it is easy to seethat K ( T ) = { ak + bk | k ∈ K ( S ) , k ∈ K ( S ) } .Since t ( T ) = t ( S ) t ( S ), it follows that T is symmetric if and only if both S and S aresymmetric, so in the next theorem we exclude this case. Theorem 4.1.
Let T be a gluing of two numerical semigroups and assume that T is not symmetric.The following are equivalent: (1) T is GAS ; (2) T is ; (3) T = h S, b N i with b ∈ S odd and S is an almost symmetric semigroup, but not symmetric. Proof. (2) ⇒ (1) True by definition.(1) ⇒ (3) Let T = h aS , bS i . Since T is not symmetric, we can assume that S is not symmetricand, then, F( S ) = k + k for some k , k ∈ K ( S ). This implies thatF( T ) − b (F( S ) + a ) = a F( S ) + b F( S ) + ab − b F( S ) − ab = ak + ak ∈ K ( T ) \ K ( T )because F( S ) + a ∈ S . Therefore, since T is GAS, F( S ) + a is a minimal generator of S . Bydefinition of gluing, a is not a minimal generator of S , so write a = s + s ′ with s , s ′ ∈ M ( S ).Since F( S ) + s + s ′ is a minimal generator of S , we get F( S ) + s = F( S ) + s ′ = 0, i.e. F( S ) = − a = s + s ′ = 2. This proves that T = h S , b N i . Clearly, b is odd by definition of gluing, sowe only need to prove that S is almost symmetric. Assume by contradiction that it is not almostsymmetric and let s ∈ M ( S ) such that F( S ) − s = k + k ∈ K ( S ) \ K ( S ) with k , k ∈ K ( S ).Then F( T ) − (2 s + b ) = 2 F( S ) − b + 2 b − s − b = 2 k + 2 k ∈ K ( T ) \ K ( T )and 2 s + b is not a minimal generator of T , contradiction.(3) ⇒ (2) Since S is not symmetric, h K ( S ) i\ K ( S ) = 2 K ( S ) \ K ( S ) = { F( S ) } . Consider an element z ∈ h K ( T ) i \ K ( T ), that is z = 2 k + bλ + · · · + 2 k r + bλ r = 2( k + · · · + k r ) + b ( λ + · · · + λ r )for some k , . . . , k r ∈ K ( S ) and λ , . . . , λ r ∈ N . Since z / ∈ K ( T ), then k + · · · + k r / ∈ K ( S ) andso k + · · · + k r = F( S ). Therefore, z = 2 F( S ) + b ( λ + · · · + λ r ) ∈ K ( T ) \ K ( T ) and, since it isnot in K ( T ) and F( T ) = 2 F( S ) + b , it follows that either z = 2 F( S ) or z = 2 F( S ) + b . Hence, |h K ( T ) i \ K ( T ) | = 2 and thus T is 2-AGL. (cid:3) Numerical Duplication.
In the previous subsection we have shown that if a non-symmetricGAS semigroup is a gluing, then it can be written as h S, b N i . This kind of gluing can be seen asa particular case of another construction, the numerical duplication , introduced in [7].Given a numerical semigroup S , a relative ideal I of S and an odd integer b ∈ S , the numericalduplication of S with respect to I and b is defined as S ✶ b I = 2 · S ∪ { · I + b } , where 2 · X = { x | x ∈ X } for every set X . This is a numerical semigroup if and only if I + I + b ⊆ S . This is alwaystrue if I is an ideal of S and, since in the rest of the subsection I will always be an ideal, we ignorethis condition. In this case, if S and I are minimally generated by { s , . . . , s ν } and { i , . . . , i µ } respectively, then S ✶ b I = h s , . . . , s ν , i + b, . . . , i µ + b i and these generators are minimal. Itfollows that h S, b N i = S ✶ b S . Remark 4.2.
The Frobenius number of S ✶ b I is equal to 2 F( I ) + b . Moreover, the odd pseudo-Frobenius numbers of S ✶ b I are { λ + b | λ ∈ PF( I ) } , whereas the even elements in PF( S ✶ b I )are exactly the doubles of the elements in (( M − M ) ∩ ( I − I )) \ S ; see the proof of [7, Proposition3.5]. In particular, if 2 f ∈ PF( S ✶ b I ), then f ∈ PF( S ).In this subsection we write K in place of K ( S ). We note that S −h K i ⊆ S and F( S −h K i ) = F( S ). Lemma 4.3.
Let S be a numerical semigroup, b ∈ S be an odd integer, I be an ideal of S with F( I ) = F( S ) and T = S ✶ b I . The following hold: (1) If k ∈ K , then both k and k + b are in K ( T ) . In particular, if F( S ) − x ∈ iK \ K , then F( T ) − x ∈ iK ( T ) \ K ( T ) ; (2) Let k ∈ K ( T ) . If k is odd, then k − b ∈ K , otherwise F( S ) − k / ∈ I ; (3) If I = S − h K i and k ∈ K ( T ) is even, then k ∈ jK for some j ≥ . (4) Let I = S − h K i . If F( T ) − i − b ∈ h K ( T ) i \ K ( T ) , then F( S ) − i ∈ h K i \ K for every i ∈ I . Moreover, F( S ) − x ∈ h K i \ K if and only if F( T ) − x ∈ h K ( T ) i \ K ( T ) . LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 15
Proof. (1) If k ∈ K , then 2 k + b ∈ K ( T ), since F( T ) − (2 k + b ) = 2(F( S ) − k ) / ∈ · S . Moreover,F( T ) − k = 2(F( S ) − k ) + b and F( S ) − k / ∈ I because it is not in S , so 2 k ∈ K ( T ). Therefore, ifF( S ) − x = k + · · · + k i ∈ iK \ K with k , . . . , k i ∈ K , then F( T ) − x = 2 k + · · · +2 k i − +(2 k i + b ) ∈ iK ( T ) and, clearly, it is not in K ( T ), since 2 x ∈ T .(2) Let k be odd. Since 2(F( S ) − k − b ) = 2 F( S )+ b − k = F( T ) − k / ∈ T , it follows that F( S ) − k − b / ∈ S ,i.e. k − b ∈ K . If k is even, then 2(F( S ) − k ) + b = F( T ) − k / ∈ T and, thus, F( S ) − k / ∈ I .(3) Since F( S ) − k / ∈ S − h K i by (2), there exist i ≥ a ∈ iK such that F( S ) − k + a / ∈ S ,that is k − a ∈ K . Hence, k = a + ( k − a ) ∈ ( i + 1) K .(4) If F( T ) − i − b = k + · · · + k j + . . . k n ∈ h K ( T ) i \ K ( T ) with k , . . . , k j ∈ K ( T ) even and k j +1 , . . . , k n ∈ K ( T ) odd, then F( S ) − i = k + · · · + k j + k j +1 − b + · · · + k n − b + ( n − j )2 b ∈ h K i \ K by (2) and (3). Using (1) the other statement is analogous. (cid:3) Example 4.4. 1.
In the previous lemma we cannot remove the hypothesis F( I ) = F( S ). Forinstance, consider S = h , , i , I = h , i and T = S ✶ I . Then, F( I ) = 11 = 8 = F( S ) and wehave F( S ) − ∈ K \ K , but F( T ) − / ∈ h K ( T ) i . In the third statement of the previous lemma, j may be bigger than 1. For instance, consider S = h , , , i and T = S ✶ ( S − h K i ) = h , , , , , , , , , i . Then88 , , , ∈ K ( T ), while 44 , , ∈ K \ K and 85 ∈ K \ K . Corollary 4.5.
Let b ∈ S be odd and let I = S − h K i . The following hold: (1) If S is not almost symmetric, then S ✶ b M is not GAS ; (2) S is n- AGL if and only if S ✶ b I is n- AGL .Proof. (1) Let T = S ✶ b M and let x = 0 be such that F( S ) − x ∈ K \ K . By Lemma 4.3 (1),F( T ) − x and F( T ) − (2 x + b ) are in 2 K ( T ) \ K ( T ). Even though 2 x + b and 2 x are minimalgenerators, their difference b is a pseudo-Frobenius number of T by Remark 4.2, because 0 ∈ PF( M ),hence T is not GAS.(2) Let T = S ✶ b I . By Lemma 4.3 (4) we have that F( S ) − x ∈ h K i \ K if and only if F( T ) − x ∈h K ( T ) i \ K ( T ). Moreover, if F( T ) − (2 i + b ) ∈ h K ( T ) i \ K ( T ), Lemma 4.3 (4) implies thatF( S ) − i ∈ h K i and, since i ∈ ( S − h K i ), it follows that F( S ) ∈ S , that is a contradiction. Hence, S is n -AGL if and only if T is n -AGL. (cid:3) Remark 4.6. If S is almost symmetric with type t , then M = K − ( M − M ) and, consequently, S ✶ b M is almost symmetric with type 2 t + 1 by [7, Theorem 4.3 and Proposition 4.8].If R is a one-dimensional Cohen-Macaulay local ring with a canonical module ω such that R ⊆ ω ⊆ R , in [6, Theorem 4.2] it is proved that the idealization R ⋉ ( R : R [ ω ]) is 2-AGL if andonly if R is 2-AGL. The numerical duplication may be considered the analogous of the idealizationin the numerical semigroup case, since they are both members of a family of rings that sharemany properties (see [2]); therefore, Corollary 4.5 (2) should not be surprising. In the followingproposition we generalize this result for the GAS property. Theorem 4.7.
Let S be a numerical semigroup, let b ∈ S be an odd integer and let I = S − h K i .The semigroup T = S ✶ b I is GAS if and only if S is GAS .Proof.
Assume that T is GAS and let F( S ) − x ∈ K \ K . By Lemma 4.3, F( T ) − x ∈ K ( T ) \ K ( T ),so 2 x is a minimal generator of T and, thus, x is a minimal generator of S . Now let F( S ) − x ,F( S ) − y ∈ K \ K and assume by contradiction that x − y ∈ PF( S ). In particular, S is not symmetricand, then, I = M − h K i . Moreover, F( T ) − x and F( T ) − y are in 2 K ( T ) \ K ( T ). We also noticethat x − y ∈ I − I , indeed, if i ∈ I and a ∈ h K i , it follows that ( x − y ) + i + a ∈ ( x − y ) + M ⊆ S .Therefore, Remark 4.2 implies that 2( x − y ) ∈ PF( T ); contradiction. Conversely, assume that S is GAS and let F( T ) − z = k + k ∈ K ( T ) \ K ( T ) with k , k ∈ K ( T ). If z = 2 i + b is odd and both k and k are odd, then i ∈ I and F( S ) − i =( k − b ) / k − b ) / b ∈ K by Lemma 4.3.(2); on the other hand, if k and k are both even,F( S ) − i = k / k / ∈ h K i by Lemma 4.3.3. Since i ∈ ( S − h K i ), in both cases we get F( S ) ∈ S ,that is a contradiction. Hence, z = 2 x is even. If k is even and k is odd, Lemma 4.3 impliesthat F( S ) − x = k / k − b ) / ∈ ( j + 1) K \ K for some j ≥ x is a minimal generator of S , i.e. z = 2 x is a minimal generator of T . Moreover,let F( T ) − x , F( T ) − y ∈ K ( T ) \ K ( T ) and assume by contradiction that 2 x − y ∈ PF( T ).Remark 4.2 implies that x − y ∈ PF( S ) ⊆ K ∪ { F( S ) } . Thus, if F( T ) − x = k + k with k , k ∈ K ( T ) and k even, then F( S ) − x = k / k − b ) / ∈ h K ( S ) i \ K ( S ) by Lemma 4.3 and, so,F( S ) − y = k / k − b ) / x − y ) ∈ h K ( S ) i \ K ( S ). Hence, Theorem 3.7 yields a contradiction,because x − y ∈ PF( S ). (cid:3) Example 4.8. 1.
Consider the semigroup S in Example 4.4.2. It is GAS and, then, the previoustheorem implies that also T = S ✶ ( S − h K i ) is GAS. However, we notice that 2 K \ K = { , , } , 3 K \ K = { } and 4 K = 3 K , while 2 K ( T ) \ K ( T ) = { , , , } and2 K ( T ) = 3 K ( T ). Despite Theorem 4.7, if S ✶ b I is GAS for an ideal I different form S − h K i , it is not true thatalso S is GAS. For instance, the semigroup S in Example 4.4.1 is not GAS, but S ✶ I is.4.3. Dilatations of numerical semigroups.
We complete this section studying the transfer ofthe GAS property in a construction recently introduced in [4]: given a ∈ M − M , the numericalsemigroup S + a = { } ∪ { m + a | m ∈ M } is called dilatation of S with respect to a . Proposition 4.9.
Let a ∈ M − M . The semigroup S + a is GAS if and only if S is GAS .Proof.
We denote the semigroup S + a by T . Recalling that F( T ) = F( S ) + a , by [4, Lemma 3.1and Lemma 3.4] follows that 2 K ( T ) = 2 K ( S ) and2 K ( S ) \ K ( S ) = { F( S ) − x , . . . , F( S ) − x r , F( S ) } , K ( T ) \ K ( T ) = { F( T ) − ( x + a ) , . . . , F( T ) − ( x r + a ) , F( T ) } for some x , . . . , x r ∈ M .Assume that S is a GAS semigroup. Then, x i is a minimal generator of S and it is straightforwardto see that x i + a is a minimal generator of T . Moreover, if ( x i + a ) − ( x j + a ) ∈ PF( T ), then forevery m ∈ M we have x i − x j + m + a ∈ T , i.e. x i − x j + m ∈ M , that is a contradiction, since S is GAS.Now assume that T is GAS. Suppose by contradiction that x i is not a minimal generator of S , that is x i = s + s for some s , s ∈ M . We have F( S ) − ( s + s ) ∈ K ( S ) \ K ( S ) andso F( S ) − s ∈ K ( S ) \ K ( S ), since 2 K ( S ) is a relative ideal. Hence, s = x j for some j and( x i + a ) − ( x j + a ) = s ∈ S . Since x i + a is a minimal generator, we have that s / ∈ T . Moreover,for every m + a ∈ M ( T ) we clearly have s + m + a ∈ M ( T ), because s ∈ S . This yields acontradiction because ( x i + a ) − ( x j + a ) = s ∈ PF( T ) and T is GAS. Finally, if x i − x j ∈ PF( S ),it is trivial to see that x i − x j ∈ PF( T ). (cid:3) Remark 4.10.
Suppose 2 K ( S + a ) \ K ( S + a ) = { F( S + a ) − ( x + a ) , . . . , F( S + a ) − ( x r + a ) , F( S + a ) } with x + a, . . . , x r + a minimal generators of S + a , but S + a is not GAS. Then 2 K ( S ) \ K ( S ) = { F( S ) − x , . . . , F( S ) − x r , F( S ) } , but it is not necessarily true that x , . . . , x r are minimal generatorsof S . For instance, consider S = h , , i and S + 7 = h , , , , , , , , , i . In thiscase 2 K ( S + 7) \ K ( S + 7) = { − , − , } and 2 K ( S ) \ K ( S ) = { − , − , } . LMOST CANONICAL IDEALS AND GAS NUMERICAL SEMIGROUPS 17
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Marco D’Anna - Dipartimento di Matematica e Informatica - Universit`a degli Studi di Catania -Viale Andrea Doria 6 - 95125 Catania - Italy
E-mail address : [email protected] Francesco Strazzanti - Dipartimento di Matematica e Informatica - Universit`a degli Studi di Cata-nia - Viale Andrea Doria 6 - 95125 Catania - Italy
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