On pseudo-Frobenius elements of submonoids of N^d

Abstract

In this paper we study those submonoids of \(\mathbb {N}^d\) with a nontrivial 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 \(\mathbb {N}^d\) and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.

Full text

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+bawhich 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−bfand 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 maximalsSAp(S, b)6=∅. In this case,
(1) PF(S) = {a−b|a∈maximalsSAp(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∈maximalsSAp(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∈maximalsSAp(S, b). Consider now a00 ∈
maximalsSAp(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∈maximalsSSi\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) := infNd(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) = minNd(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= minNd(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) = minNd(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: xyif 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}= maximalsS(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) = maximalsS(Ap(S, m(S)) −m(S). Therefore, Ap(S, m(S)) \ {0}=
maximalsS(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, maximalsS(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