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ON PSEUDO-FROBENIUS ELEMENTS OF SUBMONOIDS

OF Nd

J.I. GARC´

IA-GARC´

IA, I. OJEDA, J.C. ROSALES, AND A. VIGNERON-TENORIO

Abstract. In this paper we study those submonoids of Ndwhich a non-

trivial pseudo-Frobenius set. In the aﬃne case, we prove that they are

the aﬃne semigroups whose associated algebra over a ﬁeld has maximal

projective dimension possible. We prove that these semigroups are a

natural generalization of numerical semigroups and, consequently, most

of their invariants can be generalized. In the last section we introduce

a new family of submonoids of Ndand using its pseudo-Frobenius ele-

ments we prove that the elements in the family are direct limits of aﬃne

semigroups.

Introduction

Throughout this paper Nwill denote the set of nonnegative integers. Un-

less otherwise stated, all considered semigroups will be submonoids of Nd.

Finetelly generated submonoids of Ndwill be called, aﬃne semigroups, as

ussual. If d= 1, aﬃne semigroups are called numerical semigroups. Numer-

ical semigroup has been widely studied in the literature (see, for instance,

[13] and the references therein). It is well-known that S⊆Nis a numerical

semigroup if and only if Sis a submonoid of Nsuch that N\Sis a ﬁnite set.

Clearly, this does not hold for aﬃne semigroups in general. In [8], the aﬃne

semigroups whose complementary in the cone that they generate is ﬁnite,

in a suitable sense, are called C−semigroups and the authors prove that a

Wilf’s conjecture can be generalized to this new situation.

In the numerical case, the ﬁniteness of N\Simplies that there exists at

least a positive integers a∈N\Ssuch that a+S\{0} ⊆ S(provided that S6=

N). These integers are called pseudo-Frobenius numbers and the biggest one

Date: March 27, 2019.

2010 Mathematics Subject Classiﬁcation. 20M14, 13D02.

Key words and phrases. Aﬃne semigroups, numerical semigroups, Frobenius elements,

pseudo-Frobenius elements, Ap´ery sets, Gluings, Free resolution, irreducible semigroups.

The ﬁrst and the third author were partially supported by Junta de Andaluc´ıa research

group FQM-366 and by the project MTM2017-84890-P.

The second author was partially supported by the research groups FQM-024

(Junta de Extremadura/FEDER funds) and by the project MTM2015-65764-C3-1-P

(MINECO/FEDER, UE) and by the project MTM2017-84890-P.

The fourth author was partially supported by Junta de Andaluc´ıa research group FQM-

366, by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE) and by the project

MTM2017-84890-P.

1

arXiv:1903.11028v1 [math.AC] 26 Mar 2019

2 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

is the so-called Frobenius number. In this paper, we consider the semigroups

such that there exists at least one element a∈Nd\Swith a+S\ {0} ⊆ S.

By analogy, we call these elements the pseudo-Frobenius elements of S. We

emphasize that the semigroups in the family of C−semigroups have pseudo-

Frobenius elements.

One the main results in this paper is Theorem 6 which states that an

aﬃne semigroup, S, has pseudo-Frobenius elements if and only if the length

of the minimal free resolution of the semigroup algebra k[S],with kbeing

a ﬁeld, as a module over a polynomial ring is maximal, that is, if k[S] has

the maximal projective dimension possible (see Section 2). For this reason

we will say that aﬃne semigroups with pseudo-Frobenius elements are max-

imal projective dimension semigroups, MPD-semigroups for short. Observe

that, as minimal free resolution of semigroup algebras can be eﬀectively

computed, our results provides a computable necessary and suﬃcient con-

dition for an aﬃne semigroup to be a MPD-semigroup. In fact, using the

procedure outlined [12], one could theoretically compute the minimal free

resolution of k[S] from the pseudo-Frobenius elements. To this end, the ef-

fective computation of the pseudo-Frobenius is needed. We provide a bound

of the pseudo-Frobenius elements in terms of dand the cardinal and size of

the minimal generating set of S(see Corollary 9).

In order to emulate the maximal property of the Frobenius element of

a numerical semigroup, we ﬁx a term order on Ndand deﬁne Frobenius

elements as the maximal elements for the ﬁxed order of the elements in

the integral points in the complementary of an aﬃne semigroup Swith

respect to its cone. These Frobenius elements (if exist) are necesarily pseudo-

Frobenius elements of Sfor a maximality matter (see Lemma 12). So, if

Frobenius elements of Sexist then k[S] has maximal projective dimension,

unfortunately the converse it is not true in general. However, there are

relevant families of aﬃne semigroups with Frobenius elements as the family

of C−semigroups. In the section devoted to Frobenius elements (Section

3), we prove generalizations of well-known results for numerical semigroups

such as Selmer’s Theorem that relates the Frobenius numbers and Ap´ery

sets (Theorem 16) or the characterization of the pseudo Frobenius elements

in terms of the Ap´ery sets (Proposition 17). We close this section proving

that aﬃne semigroups having Frobenius elements are stable by gluing, by

giving a formula for a Frobenius element in the gluing (Theorem 18).

In the Section 4, we deal with the problem of the irreducibility of the

MPD-semigroups. Again, we prove the analogous results for MPD-semigroups

than the known-ones for irreducible numerical semigroups. Of special inter-

est is the characterization of the pseudo-Frobenius sets for C−semigroups

(Theorem 22 and Proposition 25). Pedro A. Garc´ıa-S´anchez communicated

us that similar results were obtained by C. Cisto, G. Fiolla, C. Peterson

and R. Utano in [4] for Nd−semigroups, that is, those C−semigroups whose

associated cone is the whole Nd.

On pseudo-Frobenius elements of submonoids of Nd3

Finally, in the last section of this paper, we introduce a new family of

(non-necessarily ﬁnitely generated) submonoids of Ndwhich have pseudo-

Frobenius elements, the elements of the family are called PI-monoids. These

monoids are a natural generalization of the MED-semigroups (see [13, Chap-

ter 3]). We conclude the paper by proving that any PI-monoid is direct limit

of MPD-semigroups.

The study of Nd−semigroups, which is a subfamily of the MPD-semigroups,

is becoming to be an active research area in aﬃne semigroups theory. For

instance, in [3] algorithms for dealing with Nd−semigroups are given. These

algorithms are implemented in the development version site of the GAP

([11]) NumericalSgps package ([10]):

https://github.com/gap-packages/numericalsgps.

1. Pseudo-Frobenius elements of affine semigroups

Set A={a1,...,an} ⊂ Ndand let Sbe the submonoid of Ndgenerated

by A. Consider the cone of Sin Qd

≥0

pos(S) := (n

X

i=1

λiai|λi∈Q≥0, i = 1, . . . , n)

and deﬁne H(S) := (pos(S)\S)∩Nd.

Deﬁnition 1. An integer vector a∈ H(S) is called a pseudo-Frobenius

element of Sif a+S\ {0} ⊆ S. The set of pseudo-Frobenius elements of S

is denoted by PF(S).

Observe that the set PF(S) may be empty: indeed, let

A={(2,0),(1,1),(0,2)} ⊂ N2.

The semigroup Sgenerated by Ais the subset of points in N2whose sum

of coordinates is even. Thus, we has that H(S) + S=H(S). Therefore

PF(S) = ∅.

On other hand, PF(S)6=H(S) in general as the following example shows.

Example 2. Let Sbe the submonoid of N2generated by the columns of

the following matrix

A=35012

00133

In this case,

H(S) = {(1,0),(2,0),(4,0),(1,1),(2,1),(4,1),

(1,2),(2,2),(4,2),(7,0),(7,1),(7,2)},

whereas PF(S) = {(7,2)}:

4 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

12345678

1

2

3

4

0

a1a2

a3

a4a5

v

The elements in Sare the blue points and the points in the shadowed blue

area. The red vector is the only pseudo-Frobenius element of S. The big

blue points correspond to the minimal generators of S.

There are relevant families of numerical semigroups for which the pseudo-

Frobenius element exist.

Deﬁnition 3. If H(S) is ﬁnite, then Sis said to be a C−semigroup, where

Cdenotes the cone pos(S).

The C−semigroups were introduced in [8].

Proposition 4. If Sis a C −semigroup diﬀerent from C ∩Nd, then PF(S)6=

∅.

Proof. Let be a term order on Ndand set a:= max(H(S)). If b∈S\{0}

is such that a+b6∈ S, then a+b∈ H(S) and a+bawhich contradicts

the maximality of a.

The converse of the above proposition is not true, as the following example

shows.

Example 5. Let A ⊂ N2be the columns of the matrix

A=18 18 4 20 23 8 11 11 10 14 7 7

9 3 1 8 10 3 5 2 3 3 2 3

and let Sbe the subsemigroup of N2generated by A. The elements in Sare

the integer points in an inﬁnite family of homotetic pentagons.

On pseudo-Frobenius elements of submonoids of Nd5

This semigroup is a so-called multiple convex body semigroup (see [9] for

further details). Clearly, Sis not a C−semigroup, but (13,4) ∈PF(S).

2. Maximal projective dimension

As in the previous section, set A={a1,...,an} ⊂ Ndand let Sbe the

submonoid of Ndgenerated by A. Let kbe an arbitrary ﬁeld.

The surjective k-algebra morphism

ϕ0:R:= k[x1, . . . , xn]−→ k[S] := M

a∈S

kχa;xi7−→ χai

is S-graded, thus, the ideal IS:= ker(ϕ0) is a S-homogeneous ideal called

the ideal of S. Notice that ISis a toric ideal generated by

nxu−xv:

n

X

i=1

uiai=

n

X

i=1

viaio.

Now, by using the S-graded Nakayama’s lemma recursively, we may con-

struct S−graded k−algebra homomorphism

ϕj+1 :Rsj+1 −→ Rsj,

corresponding to a choice of a minimal set of S−homogeneous generators

for each module of syzygies Nj:= ker(ϕj), j ≥0 (see [5] and the references

therein). Notice that N0=IS.Thus, we obtain a minimal free S−graded

resolution for the R−module k[S] of the form

. . . −→ Rsj+1 ϕj+1

−→ Rsj−→ . . . −→ Rs2ϕ2

−→ Rs1ϕ1

−→ Rϕ0

−→ k[S]−→ 0,

where sj+1 := Pb∈SdimkVj(b),with Vj(b) := (Nj)b/(mNj)b,is the so-

called (j+1)th Betti number of k[S], where m=hx1, . . . , xniis the irrelevant

maximal ideal.

Observe that the dimension of Vj(b) is the number of generators of degree

bin a minimal system of generators of the jth module of syzygies Nj(i.e. the

multigraded Betti number sj,b). The b∈Ssuch that Vj(b)6= 0 are called

S−degrees of the j−minimal syzygy of k[S]. So, by the Noetherian

property of R, sj+1 is ﬁnite. Moreover, by the Hilbert’s syzygy theorem

and the Auslander-Buchsbaum’s formula, it follows that sj= 0 for j >

p=n−depthRk[S] and sp6= 0. Such integer pis called the projective

dimension of S.

Since depthRk[S]≥1, the projective dimension of Sis lesser than or

equal to n−1. We will say that Sis a maximal projective dimension

semigroup (MPD-semigroup, for short) if its projective dimension is n−1,

equivalently, if depthRk[S] = 1.

Recall that Sis said to be Cohen-Macaulay if depthRk[S] = dim(k[S]).

So, if Sis a MPD-semigroup, then Sis Cohen-Macaulay if and only if k[S]

is the coordinate ring of a monomial curve; equivalently, Sis a numerical

semigroup.

6 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

Theorem 6. The necessary and suﬃcient condition for Sto be a MPD-

semigroup is that PF(S)6=∅. In this case, PF(S)has ﬁnite cardinality.

Proof. By deﬁnition, Sis a MPD-semigroup if and only if Vn−2(b)6=∅for

some b∈S. By [5, Theorem 2.1], given b∈S,Vn−2(b)6=∅if and only if

b−Pn

i=1 ai6∈ Sand b−Pi∈Fai∈S, for every F({1, . . . , n}. Clearly, if

PF(S)6=∅, there exists a∈ H(S) such that a+S\ {0} ⊆ S. So, by taking

b=a+Pn

i=1 ai, one has that b−Pn

i=1 ai6∈ Sand b−Pi∈Fai∈S, for every

F({1, . . . , n}, and we conclude Sis a MPD-semigroup. Conversely, if there

exists b∈Ssuch that Vn−2(b)6=∅, then we have that a:= b−Pn

i=1 ai∈

Zd\S. Clearly, a∈Zd\Sand a+S\ {0} ⊆ S, because a+aj∈S, for

every j∈ {1, . . . , n}.Thus, in order to see that a∈ H(S) it suﬃces to prove

that a∈pos(S). Without loss of generality, we assume that {a1, . . . , a`}is a

minimal set of generators of pos(S). By Farkas’ Lemma, pos(S) is a rational

convex polyhedral cone. Then, for each j∈ {1, . . . , `}there exists cj∈Rd

such that aj·cj= 0 and ai·cj≥0, i 6=j, where ·denotes the usual inner

product on Rn. Now, since a+aj∈S, one has that a+aj=Pn

i=1 uij ai,

for some uij ∈N. Therefore

a·cj= (a+aj)·cj= (

n

X

i=1

uij ai)·cj=

n

X

i=1

uij (ai·cj)≥0,

for every j∈ {1, . . . , `}. That is to say a∈pos(S).

Finally, if PF(S)6=∅, the ﬁniteness of PF(S) follows from the ﬁniteness

of the Betti numbers.

Observe that from the arguments in the proof of Theorem 6 it follows

that spis the cardinality of PF(S). In fact, we have proved the following

fact:

Corollary 7. If Sis a MPD-semigroup, then b∈Sis the S−degree of the

(n−2)th minimal syzygy of k[S]if and only if b∈ {a+Pn

i=1 ai,a∈PF(S)}.

In [12], it is outlined a procedure for a partial computation of the minimal

free resolution of Sstarting from a set of S−degrees of the jth minimal

syzygy of k[S],for some j. This procedure gives a whole free resolution if

one knows all the S−degrees of the pth minimal syzygies of k[S], where

pis the projective dimension of S. Therefore, by Corollary 7, if Sis a

MPD-semigroup, we can use the proposed method in [12] to compute the

minimal free resolution of k[S], provided that we were able to compute

PF(S). However, this is not easy at all, for this reason it is highly interesting

to given bounds for the elements in PF(S).

Given u= (u1, . . . , un)∈Nn, let `(u) be the length of uthat is, `(u) =

Pn

i=1 ui, and for an d×n−integer matrix B= (b1|···|bn), we will write

||B||∞for maxiPn

j=1 |bij |. In [6], the authors provide an explicit bound for

the S−degrees of the minimal generators of Nj, for every j∈ {1, . . . , p}.

Let A∈Nd×nbe the matrix whose ith column is ai, i = 1, . . . , n.

On pseudo-Frobenius elements of submonoids of Nd7

Theorem 8. [6, Theorem 3.2] If b∈Sis an S−degree of a minimal

j−syzygy of k[S], then b=Auwith u∈Nnsuch that

`(u)≤(1 + 4 ||A||∞)d(dj−1) + (j+ 1)dj−1,

where dj=n

j+ 1 .

By Corollary 7, this bound can be particularized for j=n−2 as follows.

Corollary 9. Let Sbe a MPD-semigroup. If a∈PF(S), then a=Avfor

some v∈Nnsatisfying

`(v)≤(1 + 4 ||A||∞)d(n−1) +n2−1.

Proof. Assuming ais a pseudo-Frobenius element of S, by Corollary 7, there

exists b=Au∈Sfor some u∈Nn,with Vn−2(b)6=∅and such that

a=b−Pn

i=1 ai; in particular a=Au−A1=A(u−1). Consider v=u−1.

Note that ||v||1≤ ||u||1+||1||1=||u||1+n. By Theorem 8, ||u||1≤

(1 + 4||A||∞)d(dj−1) + (j+ 1)dj−1 where j=n−2 and dn−2=n. So,

`(v)≤(1 + 4 ||A||∞)d(n−1) + (n−1)n−1 + n

= (1 + 4||A||∞)d(n−1) +n2−1,

as claimed.

Note that, given any aﬃne semigroup and the graded minimal free resolu-

tion of its associated algebra over a ﬁeld, Theorem 6 and Corollary 7 allow us

to check if the semigroup has pseudo-Frobenius elements and, in aﬃrmative

case, to compute them. Thus, the combination of both results provide an

algorithm for the computation of the pseudo-Frobenius elements of a aﬃne

semigroup, provided that they exist (i.e. if the depth of the algebra is one).

The following example illustrates this fact:

Example 10. Let Sbe the multiple convex body semigroup associated to

the convex hull Pof the set {(1.2, .35),(1.4,0),(1.5,0),(1.4,1)}, that is to

say,

S=[

k∈N

kP ∩ N2.

8 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

Using the Mathematica package PolySGTools introduced in [9], we obtain

that the minimal generating system of Sis the set of columns of the following

matrix

A=3445777789

0122034512.

Now, we can easily check that Sis not a C−semigroup, because (3,2) +

λ(7,5) ∈ H(S),for every λ∈N. Moreover, we can compute the S−graded

minimal free resolution of k[S] using Singular ([7]) as follows:

LIB "toric.lib";

LIB "multigrading.lib";

ring r = 0, (x(1..10)), dp;

intmat A[2][10] = 3, 4, 4, 5, 7, 7, 7, 7, 8, 9,

0, 1, 2, 2, 0, 3, 4, 5, 1, 2;

setBaseMultigrading(A);

ideal i = toric_ideal(A,"ect");

def L = multiDegResolution(i,9,0);

Finally, using the command multiDeg(L[9]), we obtain that the degrees of

the minimal generators of the 9−th syzygy module are (72,20) and (73,21).

So, Shas two pseudo-Frobenius elements: (11,0) = (72,20) −(61,20) and

(12,1) = (73,21) −(61,20).

3. On Frobenius elements of MPD-semigroups

Throughout this section, Swill be a MPD-semigroup generated by A=

{a1,...,an} ⊂ Nd.

Deﬁnition 11. We say that f∈ H(S) is a Frobenius element of Sif

f= max≺H(S) for some term order ≺on Nd.Let us write F(S) for the set

of Frobenius elements of S.

Frobenius elements of Smay not exist. However, if Sis a C−semigroup,

then it has Frobenius elements because H(S) is ﬁnite.

Lemma 12. Every Frobenius element of Sis a pseudo-Frobenius element

of S, in symbols: F(S)⊆PF(S).

Proof. If f∈F(S), then there is a term order ≺on Ndsuch that f=

max≺H(S). If there exists a∈S\ {0}such that f+a6∈ S, then f≺f+a∈

H(S), in contradiction to the maximality of f. Therefore f∈PF(S).

The following notion of Frobenius vectors was introduced in [2]: we say

that Shas a Frobenius vector if there exists f∈G(A)\Ssuch that

f+ relint(pos(S)) ∩G(A)⊆S\ {0} ⊆ S,

where G(A) denotes the group generated by Ain Zdand relint(pos(S)) the

relative interior of the cone pos(S).

Proposition 13. Every Frobenius element of Sis a Frobenius vector of S.

On pseudo-Frobenius elements of submonoids of Nd9

Proof. Let f∈F(S). Since a:= f+a1∈S, then f=a−a1∈G(A)\S.

Let b∈relint(pos(S)) ∩G(A). If b∈S, then f+b∈Sby Lemma 12. If

b6∈ S, then b∈ H(S) = (pos(S)\S)∩Ndand therefore either f+b∈Sor

f+b∈ H(S). However, since f+bis greater than ffor every term order

on Nd, we are done.

As a consequence of the above result, we have that the set of C-semigroups

is a new family of aﬃne semigroups for which Frobenius vectors exist.

Although Frobenius vectors may not exist in general, there are families

of submonoids of Ndwith Frobenius vectors that are not MPD-semigroups

(see [2]). However, even for MPD-semigroups, the converse of the above

proposition does not hold in general, for instance, the MPD-semigroup in

Example 5 has a Frobenius vector which is not a Frobenius element.

Now, similarly to the numerical case (n= 1), we can give a Selmer’s

formula type (see [13, Proposition 2.12(a)]) for C−semigroups. To do this,

we need to recall the notion of Ap´ery set.

Deﬁnition 14. The Ap´ery set of a submonoid Sof Ndrelative to b∈

S\ {0}is deﬁned as Ap(S, b) = {a∈S|a−b∈pos(S)\S}.

Clearly Ap(S, b)−b⊆ H(S); in particular, if Sis C−semigroup, we have

that Ap(S, b) is ﬁnite for every b∈S\ {0}.

Proposition 15. Let Sbe a submonoid of Ndand b∈S\ {0}. For each

a∈Sthere exists an unique (k, c)∈N×Ap(S, b)such that a=kb+c. In

particular, Ap(S, b)\ {0}∪ {b}is a system of generators of S.

Proof. If suﬃces to take kas the highest non-negative integer such that

b−ka∈S.

The following result is the generalization of [13, Proposition 2.12(a)].

Theorem 16. If f∈F(S), there exists a term order ≺on Ndsuch that

f= max

≺Ap(S, b)−b,

for every b∈S\ {0}.

Proof. By deﬁnition, there exists a term order ≺on Ndsuch that f=

max≺H(S). By Lemma 12, f+b∈Sand clearly (f+b)−b=f6∈ pos(S)\S.

Thus, f+b∈Ap(S, b). Now, suppose that there exist a∈Ap(S, b) such

that f+b≺a. In this case, a−b∈ H(S), so a−bfand therefore

a≺f+bwhich contradicts the anti-symmetry property of ≺.

Next result generalizes [13, Proposition 2.20].

Proposition 17. Let Sbe a submonoid of Ndand b∈S\ {0}. Then

PF(S)6=∅if and only if maximalsSAp(S, b)6=∅. In this case,

(1) PF(S) = {a−b|a∈maximalsSAp(S, b)}.

10 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

Proof. Suppose that there exists b0∈PF(S) and let a=b0+b. Clearly

a∈Ap(S, b), and we claim that a∈maximalsSAp(S, b). Otherwise, there

exists a0∈Ap(S, b) such that a0−a∈S, then

a0−b= (a0−b) + b0−b0=b0+ (a0−(b0+b)) = b0+ (a0−a)∈S,

which contradicts the deﬁnition of Ap´ery set of Srelative to b. There-

fore, b0=a−bwith a∈maximalsSAp(S, b). Consider now a00 ∈

maximalsSAp(S, b) and let b00 =a00 −b. If b00 6∈ PF(S),then b00 +ai6∈ S,

for some i∈ {1, . . . , n}, that is to say, a00 +ai∈Ap(S, b) which is not

possible by the maximality of a00 in Ap(S, b) with respect to S.

Observe that (1) holds for every MPD-semigroup.

We end this section by proving that MPD-semigroups are stable by gluing.

First of all, let us recall the notion of gluing of aﬃne semigroups.

Given an aﬃne semigroup S⊆Nd, denote by G(S) the group spanned by

S, that is,

G(S) = z∈Zm|z=a−b,a,b∈S.

Assume that Sis ﬁnitely generated. Let Abe the minimal generating

system of Sand A=A1∪ A2be a nontrivial partition of A. Let Sibe the

submonoid of Ndgenerated by Ai, i ∈ {1,2}. Then S=S1+S2. We say

that Sis the gluing of S1and S2by dif

•d∈S1∩S2and,

•G(S1)∩G(S2) = dZ.

We will denote this fact by S=S1+dS2.

Theorem 18. Let Sbe an ﬁnitely generated submonoid of Nd. Assume that

S=S1+dS2. If S1and S2are MPD-semigroups, and bi∈PF(Si), i = 1,2,

then b1+b2+d∈PF(S). In particular, Sis a MPD-semigroup.

Proof. Let b:= b1+b2+d. Since bi∈pos(Si), i = 1,2,and d∈S1∩S2,

we conclude that b∈pos(S). If b∈S, then there exist b0

i∈Si, i = 1,2,

such that b=b0

1+b0

2. Then b1+d−b0

1=b0

2−b2∈G(S1)∩G(S2) = dZ.

So, there exist k∈Zsuch that b1+d−b0

1=b0

2−b2=kd. If k≤0,then

b2=b0

2−kd∈S2, which is impossible. If k > 0,then b1=b0

1+ (k−1)d∈

S1, which is also impossible. All this prove that b∈ H(S).

Now, let a∈S\ {0}. Again there exist b0

i∈Si, i = 1,2,such that

a=b0

1+b0

2. Since d∈S1∩S2⊂S. We have that b1+b0

1∈S1and

b2+b0

2+d∈S2. Thus b+a∈S, and we are done.

Example 19. Let S1={(x, y, z)∈N3|z= 0}\{(1,0,0)}and S2=

{(x, y, z)∈N3|x=y}\{(0,0,1)}. Clearly, (1,0,0) ∈PF(S1) and (0,0,1) ∈

PF(S2). They are minimally generated by {(2,0,0),(3,0,0),(0,1,0),(1,1,0)}

and {(1,1,0),(1,1,1),(0,0,2),(0,0,3)}, respectively. The set G(S1)∩G(S2)

is equal to (1,1,0)Zand S1+S2is generated by

{(2,0,0),(3,0,0),(0,1,0),(1,1,0),(1,1,1),(0,0,2),(0,0,3)}

By Theorem 18, (1,0,0)+(0,0,1)+(1,1,0) = (2,1,1) belongs to PF(S1+S2).

On pseudo-Frobenius elements of submonoids of Nd11

4. On the irreducibility of MPD-semigroups

Now, let us study the irreducibility of MPD-semigroups with special em-

phasis in the C−semigroups case. Recall that a submonoid of Ndis irre-

ducible if cannot be expressed as an intersection of two submonoids of Nn

containing it properly.

Lemma 20. Let a∈PF(S). If 2a∈S, then S∪ {a}is the submonoid of

Ndgenerated by A∪{a}. Moreover,

(a) if a∈F(S)and PF(S)6={a}then S∪ {a}is a MPD-semigroup;

(b) if Sis a C−semigroup, then S∪ {a}is a C−semigroup.

Proof. By deﬁnition, a+b∈S⊂S∪{a}, for every b∈S, and by hypothesis

2a∈S, so ka∈Sfor every k∈N; thus, S∪ {a}is the submonoid of Nd

generated by A∪{a}.

Suppose now that ais a Frobenius element of Sand that PF(S)6={a}.

Let b∈PF(S)\ {a}. Clearly, a+b∈S, because a+bis greater than a

for every term order on Nd, therefore b+ (S∪ {a})\ {0} ⊂ S⊂S∪ {a}and

we conclude that PF(S∪ {a})6=∅which proves (a).

Finally, since a∈ H(S), we have that pos(S∪ {a}) = pos(S). Therefore

H(S∪ {a})⊂ H(S), that is, S∪ {a}is a C−semigroup if Sit so as claimed

in (b).

Proposition 21. If Shas a Frobenius element, f, and is irreducible then

(a) Sis maximal among all the submonoids of Ndhaving fas a Frobenius

element.

(b) Shas an unique Frobenius element.

Proof. Suppose that Sis irreducible and let S0be a submonoid of Ndhaving

fas Frobenius element. Since, by Lemma 20, S∪ {f}is a MPD-semigroup

and S= (S∪ {f})∩S0, we conclude that S=S0. Finally, if Shas two

Frobenius elements, say f1and f2, then S= (S∪ {f1})∩(S∪ {f2}) which

contradicts the irreducibility of S.

Notice that the submonoids in condition (a) are necessarily MPD-semi-

groups; indeed the existence of a Frobenius elements in a submonoid of Nd

implies that the submonoid is a MPD-semigroup by Lemma 12.

Theorem 22. If Shas a Frobenius element, f, and is irreducible, then

either PF(S) = {f}or PF(S) = {f,f/2}.

Proof. Suppose that PF(S)6={f}. Now, since PF(S) has cardinality greater

than or equal to two, there exists a∈PF(S) diﬀerent from f. If 2a∈S,

then, by Lemma 20, S∪ {a}are S∪ {f}are a submonoid of Ndwhose

intersection is S, in contradiction with irreducibility of S. Therefore, we

may assume that 2a6∈ Swhich implies 2a∈PF(S). Then f+u= 2afor

some u∈Nd, because fis greater than or equal to bfor every term order on

12 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

Nd. Notice that 4a=f+ (f+2u)∈S; so, by Lemma 20, S∪{2a}is a MPD-

semigroup. Now, if u6= 0,then S= (S∪ {2a})∩(S∪ {f}),in contradiction

with irreducibility of S. Therefore 2a=fand we are done.

Example 23. Let Sbe the MPD-semigroup of Example 2. Let us see that

Sis irreducible. If Sis not irreducible, there exist two submonoids, S1

and S2, of N2such that S=S1∩S2. Since pos(S) = N2, we have that

pos(S1) = pos(S2) = N2and it follows that H(Si)⊆ H(S), i = 1,2. On

other hand, since (7,2) 6∈ S, then (7,2) 6∈ S1or (7,2) /∈S2. Therefore, S1

or S2is a submonoid of N2such that f∈F(S1) or f∈F(S2), respectively.

Now, by Proposition 21, we conclude that S=S1or S=S2, that is, Sis

irreducible.

If n= 1, the converse Theorem 22 is also true (see [13, Section 4.1]). Let

us see that this is also happen if we ﬁx the cone.

Deﬁnition 24. If Sis C−semigroup, we say that Sis C−irreducible if

cannot be expressed as an intersection of two ﬁnitely generated submonoids

S1and S2of Ndwith pos(S1) = pos(S2) = pos(S) containing it properly.

Proposition 25. If Sis a C−semigroup such that PF(S) = {f}or PF(S) =

{f,f/2}, then Sis C−irreducible.

Proof. Suppose there exist two ﬁnitely generated submonoids S1and S2of

Ndwith pos(S1) = pos(S2) = pos(S) such that S=S1∩S2; in particular, S1

and S2are C−semigroups. For i= 1,2,we take bi∈maximalsSSi\S. Since

Si\Sis ﬁnite, biis well-deﬁned for i= 1,2. By maximality, bi+a∈S

for every a∈S\ {0}, i = 1,2, that is to say, bi∈PF(S).Therefore,

bi=f, i = 1,2 or bi=fand fj=f/2,{i, j }={1,2}. In the ﬁrst

case, we obtaim b1=b2which is not possible because bi6∈ S, i = 1,2.

In the second case, we obtain that f∈S1∩S2=Swhich obviously is

impossible. Therefore, b1or b2does not exist and we conclude that S=S1

or S=S2.

5. PI-monoids

Let Ndbe the usual partial order in Nd, that is, a= (a1, . . . , ad)Nd

b= (b1, . . . , bd) if and only if ai≤bi, i ∈ {1, . . . , d}.

Deﬁnition 26. If Sis a submonoid of Nd, we deﬁne the multiplicity of S

as m(S) := infNd(S\ {0}).

If d= 1, the notion of multiplicity deﬁned above agrees with the notion of

multipliciy of a numerical semigroup (see [13, Section 2.2]) Let us introduce

a new family of submonoids of Nd, that we have called principal ideal

monoids, or PI−monoids for short. This family generalizes the notion of

MED-semigroups (see [13, Chapter 3] for d > 1.

Deﬁnition 27. A submonoid Sof Ndis said to be a PI-monoid if there

exist a submonoid Tof Ndand a∈T\ {0}such that S=a+T∪ {0}.

On pseudo-Frobenius elements of submonoids of Nd13

Clearly, PI-monoids are not always aﬃne semigroups, since they are not

necessarily ﬁnitely generated. We will explicitly provide a minimal generat-

ing system of any PI-monoid later on, ﬁrst let us explore some its properties.

Example 28. In N2, an example of ﬁnitely generated PI-monoid is S1=

(2,2) + h(1,1)i=h(2,2),(3,3)i.

To obtain a non-ﬁnitely generated PI-monoid of N2, consider T=N2

and a= (1,1). The PI-monoid S2= (1,1) + N2is equal to {(x, y)|x≥

1, y ≥1}∪{(0,0)}which is not a ﬁnitely generated submonoid of N2.

Lemma 29. If S⊆Ndis a PI-monoid, then m(S)∈S\ {0}.In particular,

m(S) = minNd(S\ {0}).

Proof. Since Sis a PI-monoid, there exist a submonoid Tof Ndand a∈

T\ {0}such that S=a+T∪ {0}. Clearly, a= minNd(S\ {0}).

The following result is the generalization of [13, Proposition 3.12].

Proposition 30. Let Sbe a submonoid of Nd. Then, Sis a PI-monoid if

and only if m(S)∈S\ {0}and (S\ {0})−m(S)is a submonoid of Nd.

Proof. If Sis a PI-monoid, by Lemma 29, m(S) = minNd(S\ {0}); more-

over, there is a submonoid Tof Ndsuch that S= (m(S) + T)∪ {0}. So,

(S\ {0})−m(S) = Tis a submonoid of Nd. For the converse implication, it

suﬃces to note that S= (m(S)+ T)∪ {0}and that T= (S\{0})−m(S).

Corollary 31. If S⊆Ndis a PI-monoid, then there exist an unique sub-

monoid Tof Ndand an unique a∈T\ {0}such that S= (a+T)\ {0}

Proof. It is clear that amust be equal to m(S) and that Tmust be equal

to (S\ {0})−m(S).

Remark 32. Given a submonoid Sof Nd, we will write PI(S) for the set

(a+S)∪ {0} | a∈S\ {0}.

Observe that, as an immediate consequence of Corollary 31, we have that

the set {PI(S)|Sis a submonoid of Nd}is a partition of the set of all PI-

monoids of Nd. Moreover, if Adenotes the set of all submonoids of Nd, for

some d, and Pidenotes the set of all PI-monoids of Nd, for some d, we have

an injective map

A−→ Pi;S7→ (min

lex (S\ {0}) + S)∪ {0},

where lex means the lexicographic term order on Nd.

Recall that a system of generators Aof a submonoid Aof Ndis said to be

minimal if no proper subset of Agenerates A. The following result identiﬁes

a minimal system of generators of an PI-monoid.

14 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

Proposition 33. Let Sbe a submonoid of Nd. Then Sis a PI-monoid if

and only if

Ap(S, m(S)) \ {0}∪ {m(S)}

is a minimal system of generators of S.

Proof. By Lemma 29, if Sis a PI-monoid, then m(S)∈S\ {0}Moreover,

by Proposition 15, we have that A:= Ap(S, m(S)) \ {0}∪ {m(S)}is a

system of generators of S. So, it suﬃces to prove that Ais minimal. Let us

assume the contrary, that is, there exists a∈ A such that A \ {a}generates

S. By the minimality of m(S),a6=m(S). Thus, a∈Ap(S, m(S)) \ {0}

and there exists band c∈Swith a=b+c. By Proposition 30, we know

that b−m(S) + c−m(S) = d−m(S) for some d∈S\ {0}. Therefore,

a=d+m(S)6∈ Ap(S, m(S)), which is impossible. Conversely, if Sis not

a PI-monoid, by Proposition 30, we have that (S\ {0})−m(S) is not a

submonoid of Nd. So, there exists aand b∈Ap(S, m(S)) \ {0}such that

a−m(S) + b−m(S)6∈ (S\ {0})−m(S). In particular, a+b−m(S)6∈ S

and consequently a+b∈Ap(S, m(S)).So, Ap(S, m(S)) is not a minimal

system of generators of S.

Now, we will show that PI-monoids have non-trivial inﬁnite pseudo-

Frobenius set. Recall that every submonoid Sof Nddeﬁnes a natural partial

order on Ndas follows: xyif and only if y−x∈S. As in the previous

section this partial order will be denoted as S.

Corollary 34. A submonoid Sof Ndis a PI-monoid if and only if m(S)∈

S\ {0}and Ap(S, m(S)) \ {0}=m(S) + PF(S).

Proof. If Sis a PI-monoid, then m(S)∈Sby Lemma 29 and, by Propo-

sition 33, Ap(S, m(S)) \ {0}∪ {m(S)}is a minimal system of generators

of S. Therefore, Ap(S, m(S)) \ {0}= maximalsS(Ap(S, m(S)). Now, by

Proposition 17, we are done. Conversely, let us suppose that m(S)∈Sand

that Ap(S, m(S)) \ {0}=m(S) + PF(S). By Proposition 17, we have that

PF(S) = maximalsS(Ap(S, m(S)) −m(S). Therefore, Ap(S, m(S)) \ {0}=

maximalsS(Ap(S, m(S)), that is, Ap(S, m(S))\{0}∪{m(S)}is a minimal

system of generators of S. Now, by Proposition 33, we are done.

Putting all this together, we have the following characterization of the

PI-monoids.

Theorem 35. Let Sbe a submonoid of Nd. The following conditions are

equivalent:

(1) Sis a PI-monoid.

(2) m(S)∈S\ {0}and (S\ {0})−m(S)is closed under addition.

(3) Ap(S, m(S)) \ {0}∪ {m(S)}is a minimal system of generators of

S.

(4) {m(S) + PF(S)}∪{m(S)}is a minimal system of generators of S.

On pseudo-Frobenius elements of submonoids of Nd15

Example 36. Let S1and S2be the PI-monoids of Example 28. For S1

we have m(S1) = {(2,2)}and Ap(S1,(2,2)) = {(0,0),(3,3)}obtaining that

{(2,2),(3,3)}is a system of generators of S1and that PF(S1) = {(1,1)}.

For S2,m(S2) = (1,1) and Ap(S2,(1,1)) = {(0,0)} ∪ {(x, 1) |x∈N\

{0,1}} ∪ {(1, y)|y∈N\ {0,1}}. So {(1,1)} ∪ {(x, 1) |x∈N\ {0,1}} ∪

{(1, y)|y∈N\ {0,1}} is a non-ﬁnite system of generators of S2and

PF(S2) = {(x, 0) |x∈N\ {0}} ∪ {(0, y)|y∈N\ {0}}.

Finally, our last results state the relationship between PI-monoids and

MPD-semigroups.

Corollary 37. Let Sbe a PI-monoid. Then Sis an aﬃne semigroup if and

only if Sis an MPD-semigroup. In this case, Ap(S, m(S)) is ﬁnite.

Proof. Let Sbe a ﬁnitely generated PI-monoid. By Corollary 34, PF(S)6=

∅. So, by Theorem 6, Sis a MPD-semigroup. The converse is trivial as

MPD-semigroups are aﬃne semigroups by deﬁnition. The last part is a

direct consequence of Theorem 6 and Corollary 34.

The following result was inspired by [1, Lemma 2.2]:

Corollary 38. Let Sbe a PI-monoid. Then there exist a direct system

(Sλ, iλµ)of MPD-semigroups contained in Ssuch that S= lim

−→λ∈ΛSλ, where

iλµ :Sλ→Sµis the inclusion map.

Proof. Let Λ = {λ⊂ {m(S) + PF(S)} | λis ﬁnite},partially ordered by

inclusion, and deﬁne Sλto be the aﬃne semigroup generated by λ∪{m(S)}.

Clearly, we have that Sλ⊆Sµif λ⊆µ; in this case, let iλµ :Sλ→Sµis

the inclusion map. Now, since Sλ⊆Sfor every λ∈Λ, we conclude that

S= lim

−→λ∈ΛSλby Theorem 35, because {m(S) + PF(S)}∪{m(S)}is a

minimal system of generators of S.

Finally, let us see that Sλis a MPD-semigroup for every λ∈Λ. To do

that, we ﬁrst note that m(Sλ) = m(S)∈Sλ, for every λ∈Λ. Let A=

{a1,...,an} ⊆ PF(S) and let λ={m(S) + A} ∈ Λ. Then, Ap(Sλ, m(S)) =

{0,a1,...,an}is ﬁnite, in particular, maximalsS(Ap(S, m(S)) −m(S) is a

non-empty ﬁnite set. Therefore, by Proposition 17, PF(S)6=∅, that is to

say, Sλis a MPD-semigroup.

Acknowledgement. This paper was originally motivated by a question

made of Antonio Campillo and F´elix Delgado about C−semigroups during a

talk of the fourth author at the GAS seminar of the SINGACOM research

group. The question is answered in a wider context by Corollary 7. Part of

this paper was written during a visit of the second author to the Universidad

de C´adiz (Spain) and to the IEMath-GR (Universidad de Granada, Spain),

he thanks these institutions for their warm hospitality. The authors would

like to thank Antonio Campillo, F´elix Delgado and Pedro A. Garc´ıa-S´anchez

for usseful suggestions and comments.

16 J. I. Garc´ıa-Garc´ıa, I. Ojeda, J.C. Rosales and A. Vigneron-Tenorio

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Universidad de C´

adiz

Departamento de Matem´

aticas/INDESS (Instituto Universitario para el

Desarrollo Social Sostenible)

E-mail address:ignacio.garcia@uca.es

Universidad de Extremadura

Departamento de Matem´

aticas/IMUEx

E-mail address:ojedamc@unex.es

Universidad de Granada

Departamento de ´

Algebra

E-mail address:jcrosales@ugr.es

Universidad de C´

adiz

Departamento de Matem´

aticas/INDESS (Instituto Universitario para el

Desarrollo Social Sostenible)

E-mail address:alberto.vigneron@uca.es