arXiv:1104.2835v3 [] 13 Oct 2013

Combinatorial properties and characterization of
glued semigroups
J. I. Garcı́a-Garcı́a∗
M.A. Moreno-Frı́as†
A. Vigneron-Tenorio‡

Abstract
This work focuses on the combinatorial properties of glued semigroups
and provides its combinatorial characterization. Some classical results for
affine glued semigroups are generalized and some methods to obtain glued
semigroups are developed.
Keywords: Gluing of semigroups, semigroup, semigroup ideal, simplicial complex, toric ideal.
MSC-class: 20M14 (Primary), 20M05 (Secondary).

Introduction
Let S = hn1 , . . . , nl i be a finitely generated commutative semigroup with zero
element which is reduced (i.e. S ∩ (−S) = {0}) and cancellative (if m, n, n0 ∈ S
and m+n = m+n0 then n = n0 ). Under these settings if S is torsion-free then it
is isomorphic to a subsemigroup of Np which means it is an affine semigroup (see
[11]). From now on assume that all the semigroups appearing in this work are
finitely generated, commutative, reduced and cancellative, but not necessarily
torsion-free.
Let k be a field and k[X1 , . . . , Xl ] the polynomial ring in l indeterminates.
This polynomial ring is obviously an S−graded ring (by assigning the S-degree
Pl
ni to the indeterminate Xi , the S-degree of X α = X1α1 · · · Xlαl is i=1 αi ni ∈
S). It is well known that the ideal IS generated by
(
)
l
l
X
X
α
β
X −X |
α i ni =
βi ni ⊂ k[X1 , . . . , Xl ]
i=1

i=1

∗ Departamento

de Matemáticas, Universidad de Cádiz, E-11510 Puerto Real (Cádiz,
Spain). E-mail: ignacio.garcia@uca.es. Partially supported by MTM2010-15595 and Junta de
Andalucı́a group FQM-366.
† Departamento de Matemáticas, Universidad de Cádiz, E-11510 Puerto Real (Cádiz,
Spain). E-mail: mariangeles.moreno@uca.es. Partially supported by MTM2008-06201-C02-02
and Junta de Andalucı́a group FQM-298.
‡ Departamento de Matemáticas, Universidad de Cádiz, E-11405 Jerez de la Frontera (Cádiz, Spain). E-mail: alberto.vigneron@uca.es. Partially supported by the grant
MTM2007-64704 (with the help of FEDER Program), MTM2012-36917-C03-01 and Junta
de Andalucı́a group FQM-366.

1

is an S−homogeneous binomial ideal called semigroup ideal (see [6] for details).
If S is torsion-free, the ideal obtained defines a toric variety (see [12] and the
references therein). By Nakayama’s lemma, all the minimal generating sets of
IS have the same cardinality and the S−degrees of its elements can be determinated.
The main goal of this work is to study the semigroups which result from
the gluing of others two. This concept was introduced by Rosales in [10]
and it is closely related with complete intersection ideals (see [13] and the
references therein). A semigroup S minimally generated by A1 t A2 (with
A1 = {n1 , . . . , nr } and A2 = {nr+1 , . . . , nl }) is the gluing of S1 = hA1 i and
S2 = hA2 i if there exists a set of generators ρ of IS of the form ρ = ρ1 ∪
0
ρ2 ∪ {X γ − X γ }, where ρ1 , ρ2 are generating sets of IS1 and IS2 respectively,
γ
γ0
X − X ∈ IS and the supports of γ and γ 0 verify supp (γ) ⊂ {1, . . . , r}
and supp (γ 0 ) ⊂ {r + 1, . . . , l}. Equivalently, S is the gluing of S1 and S2 if
0
IS = IS1 + IS2 + hX γ − X γ i. A semigroup is a glued semigroup when it is the
gluing of others two.
As seen, glued semigroups can be determinated by the minimal generating
sets of IS which can be studied by using combinatorial methods from certain
simplicial complexes (see [1], [4] and [7]). In this work the simplicial complexes
used are defined as follows: for any m ∈ S, set
Cm = {X α = X1α1 · · · Xlαl |

l
X

αi ni = m}

(1)

i=1

and the simplicial complex
∇m = {F ⊆ Cm | gcd(F ) 6= 1},

(2)

with gcd(F ) the greatest common divisor of the monomials in F.
Furthermore, some methods which require linear algebra and integer programming are given to obtain examples of glued semigroups.
The content of this work is organized as follows. Section 1 presents the tools
to generalize to non torsion-free semigroups a classical characterization of affine
gluing semigroups (Proposition 2). In Section 2, the non-connected simplicial
complexes ∇m associated to glued semigroups are studied. By using the vertices of the connected components of these complexes we give a combinatorial
characterization of glued semigroups as well as their glued degrees (Theorem
6). Besides, in Corollary 7 we deduce the conditions in order the ideal of a
glued semigroup to be uniquely generated. Finally, Section 3 is devoted to the
construction of glued semigroups (Corollary 10) and affine glued semigroups
(Subsection 3.1).

1

Preliminaries and generalizations about glued
semigroups

A binomial of IS is called indispensable if it is an element of all system of generators of IS (up to a scalar multiple). This kind of binomials were introduced in
[9] and they have an important role in Algebraic Statistics. In [8] the authors
characterize indispensable binomials by using simplicial complexes ∇m . Note
2

that if IS is generated by its indispensable binomials, it is uniquely generated
up to a scalar multiples.
With the above notation, the semigroup S is associated
Pl to the lattice ker S
formed by the elements α = (α1 , . . . , αl ) ∈ Zl such that i=1 αi ni = 0. Given
G a system of generators of IS , the lattice ker S is generated by the elements
α − β with X α − X β ∈ G and ker S also verifies that ker S ∩ Nl = {0} if
and only if S is reduced. If M(IS ) is a minimal generating set of IS , denote
by M(IS )m ⊂ M(IS ) the set of elements whose S−degree is equal to m ∈ S
and by Betti(S) the set of the S−degrees of the elements of M(IS ). When IS
is minimally generated by rank(ker S) elements, the semigroup S is called a
complete intersection semigroup.
Let C(∇m ) be the number of connected components of ∇m . The cardinality
of M(IS )m is equal to C(∇m ) − 1 (see Remark 2.6 in [1] and Theorem 3 and
Corollary 4 in [7]) and the complexes associated to the elements in Betti(S) are
non-connected.
Construction 1. ([4, Proposition 1]). For each m ∈ Betti(S) the set M(IS )m
is obtained by taking C(∇m ) − 1 binomials with monomials in different connected components of ∇m satisfying that two different binomials have not their
corresponding monomials in the same components and fulfilling that there is at
least a monomial of every connected component of ∇m . This let us construct a
minimal generating set of IS in a combinatorial way.
Let S be minimally1 generated by A1 t A2 with A1 = {a1 , . . . , ar } and
A2 = {b1 , . . . , bt }. From now on, identify the sets A1 and A2 with the matrices
(a1 | · · · |ar ) and (b1 | · · · |bt ). Denote by k[A1 ] and k[A2 ] the polynomial rings
k[X1 , . . . , Xr ] and k[Y1 , . . . , Yt ], respectively. A monomial is a pure monomial
if it has indeterminates only in X1 , . . . , Xr or only in Y1 , . . . , Yt , otherwise it is
a mixed monomial. If S is the gluing of S1 = hA1 i and S2 = hA2 i, then the
binomial X γX −Y γY ∈ IS is a glued binomial if M(IS1 )∪M(IS2 )∪{X γX −Y γY }
is a generating set of IS and in this case the element d = S-degree(X γX ) ∈ S is
called a glued degree.
It is clear that if S is a glued semigroup, the lattice ker S has a basis of the
form
{L1 , L2 , (γX , −γY )} ⊂ Zr+t ,
(3)
where the supports of the elements in L1 are in {1, . . . , r}, the supports of the
elements in L2 are in {r + 1, . . . , r + t}, ker Si = hLi i (i = 1, 2) by considering
only the coordinates in {1, . . . , r} or {r+1, . . . , r+t} of Li , and (γX , γY ) ∈ Nr+t .
Moreover, since S is reduced, one has that hL1 i ∩ Nr+t = hL2 i ∩ Nr+t = {0}.
Denote by {ρ1i }i the elements in L1 and by {ρ2i }i the elements in L2 .
The following Proposition generalizes [10, Theorem 1.4] to non-torsion free
semigroups.
Proposition 2. The semigroup S is the gluing of S1 and S2 if and only if there
exists d ∈ (S1 ∩ S2 ) \ {0} such that G(S1 ) ∩ G(S2 ) = dZ, where G(S1 ), G(S2 )
and dZ are the associated commutative groups of S1 , S2 and {d}, respectively.
1 We consider a minimal generator set of S, in the other case S is trivially the gluing of the
semigroup generated by one of its non minimal generators and the semigroup generated by
the others.

3

Proof. Assume that S is the gluing of S1 and S2 . In this case, ker S is generated
by the set (3). Since (γX , −γY ) ∈ ker S, the element d is equal to A1 γX =
A2 γY ∈ S and d ∈ S1 ∩ S2 ⊂ G(S1 ) ∩ G(S2 ). Let d0 be in G(S1 ) ∩ G(S2 ),
then there exists (δ1 , δ2 ) ∈ Zr × Zt such that d0 = A1 δ1 = A2 δ2 . Therefore
(δ1 , −δ2 ) ∈ ker S because (A1 |A2 )(δ1 , −δ2 ) = 0 and so there exist λ, λρi 1 , λρi 2 ∈ Z
satisfying

P ρ1
 (δ1 , 0) =
i λi ρ1i + λ(γX , 0)


(0, δ2 )

−

=

P

i

λρi 2 ρ2i + λ(0, γY ),

P
and d0 = A1 δ1 = i λρi 1 (A1 |0)ρ1i + λA1 γX = λd. We conclude that G(S1 ) ∩
G(S2 ) = dZ with d ∈ S1 ∩ S2 .
Conversely, suppose there exists d ∈ (S1 ∩S2 )\{0} such that G(S1 )∩G(S2 ) =
dZ. We see IS = IS1 +IS2 +hX γX −Y γY i. Trivially, IS1 +IS2 +hX γX −Y γY i ⊂ IS .
Let X α Y β −X γ Y δ be a binomial in IS . Its S−degree is A1 α+A2 β = A1 γ +A2 δ.
Using A1 (α − γ) = A2 (β − δ) ∈ G(S1 ) ∩ G(S2 ) = dZ, there exists λ ∈ Z such
that A1 α = A1 γ + λd and A2 δ = A2 β + λd. We have the following cases:
• If λ = 0,
X αY β − X γ Y δ = X αY β − X γ Y β + X γ Y β − X γ Y δ =
= Y β (X α − X γ ) + X γ (Y β − Y δ ) ∈ IS1 + IS2 .
• If λ > 0,
X αY β − X γ Y δ =
= X α Y β −X γ X λγX Y β +X γ X λγX Y β −X γ X λγY Y β +X γ X λγY Y β −X γ Y δ =
= Y β (X α − X γ X λγX ) + X γ Y β (X λγX − Y λγY ) + X γ (Y λγY Y β − Y δ ).
Using that
X

λγX

−Y

λγY

= (X

γX

−Y

γY

)

λ−1
X

!
X

(λ−1−i)γX

Y

iγY

,

i=0

the binomial X α Y β − X γ Y δ belongs to IS1 + IS2 + hX γX − Y γY i.
• The case λ < 0 is solved similarly.
We conclude that IS = IS1 + IS2 + hX γX − Y γY i.
From above proof it is deduced that given the partition of the system of
generators of S the glued degree is unique.

2

Glued semigroups and combinatorics

Glued semigroups by means of non-connected simplicial complexes are characterized. For any m ∈ S, redefine Cm from (1), as
Cm = {X α Y β = X1α1 · · · Xrαr Y1β1 · · · Ytβt |

r
X
i=1

4

αi ai +

t
X
i=1

βi bi = m}

and consider the sets of vertices and the simplicial complexes
A1
Cm
= {X1α1 · · · Xrαr |

r
X

A1
1
αi ai = m}, ∇A
m = {F ⊆ Cm | gcd(F ) 6= 1},

i=1

A2
Cm
= {Y1β1 · · · Ytβt |

t
X

A2
2
βi bi = m}, ∇A
m = {F ⊆ Cm | gcd(F ) 6= 1},

i=1

where A1 = {a1 , . . . , ar } and A2 = {b1 , . . . , bt } as in Section 1. Trivially, the
A2
1
relations between ∇A
m , ∇m and ∇m are
A1
A2
A2
1
∇A
m = {F ∈ ∇m |F ⊂ Cm }, ∇m = {F ∈ ∇m |F ⊂ Cm }.

(4)

The following result shows an important property of the simplicial complexes
associated to glued semigroups.
Lemma 3. Let S be the gluing of S1 and S2 , and m ∈ Betti(S). Then all the
connected components of ∇m have at least a pure monomial. In addition, all
mixed monomials of ∇m are in the same connected component.
Proof. Suppose that there exists C, a connected component of ∇m only with
mixed monomials. By Construction 1 in all generating sets of IS there is at
least a binomial with a mixed monomial, but this does not occur in M(IS1 ) ∪
M(IS2 ) ∪ {X γX − Y γY } with X γX − Y γY a glued binomial.
Since S is a glued semigroup, ker S has a system of generators as (3). Let
X α Y β , X γ Y δ ∈ Cm be two monomials such that gcd(X α Y β , X γ Y δ ) = 1. In
this case, (α, β) − (γ, δ) ∈ ker S and there exist λ, λρi 1 , λρi 2 ∈ Z satisfying:

P ρ1
 (α − γ, 0) =
i λi ρ1i + λ(γX , 0)


(0, β − δ)

=

P

i

λρi 2 ρ2i − λ(0, γY )

• If λ = 0, α − γ ∈ ker S1 and β − δ ∈ ker S2 , then A1 α = A1 γ, A2 β = A2 δ
and X α Y δ ∈ Cm .
P
• If λ > 0, (α, 0) = i λρi 1 ρ1i + λ(γX , 0) + (γ, 0) and
X ρ
A1 α =
λi 1 (A1 |0)ρ1i + λA1 γX + A1 γ = λd + A1 γ,
i

then X λγX X γ Y β ∈ Cm .
• The case λ < 0 is solved likewise.
In any case, X α Y β and X γ Y δ are in the same connected component of ∇m .
We now describe the simplicial complexes that correspond to the S−degrees
which are multiples of the glued degree.
Lemma 4. Let S be the gluing of S1 and S2 , d ∈ S the glued degree and
d0 ∈ S \ {d}. Then CdA0 1 6= ∅ =
6 CdA0 2 if and only if d0 ∈ (dN) \ {0}. Furthermore,
the simplicial complex ∇d0 has at least one connected component with elements
in CdA0 1 and CdA0 2 .
5

Pr
Pt
Proof. If there exist X α , Y β ∈ Cd0 , then d0 = i=1 αi ai = i=1 βi bi ∈ S1 ∩S2 ⊂
G(S1 ) ∩ G(S2 ) = dZ. Hence, d0 ∈ dN.
Conversely, let d0 = jd with j ∈ N and j > 1 and take X γX − Y γY ∈
IS be a glued binomial. It is easy to see that X jγX , Y jγY ∈ Cd0 and thus
{X jγX , X (j−1)γX Y γY } and {X (j−1)γX Y γY , Y jγY } belong to ∇d0 .
The following Lemma is a combinatorial version of [5, Lemma 9] and it is a
necessary condition of Theorem 6.
Lemma 5. Let S be the gluing of S1 and S2 , and d ∈ S the glued degree. Then
the elements of Cd are pure monomials and d ∈ Betti(S).
Proof. The order S defined by m0 S m if m − m0 ∈ S is a partial order on S.
Assume there exists a mixed monomial T ∈ Cd . By Lemma 3, there exists
a pure monomial Y b in Cd such that {T, Y b } ∈ ∇d (the proof is analogous
if we consider X a with {T, X a } ∈ ∇d ). Now take T1 = gcd(T, Y b )−1 T and
Y b1 = gcd(T, Y b )−1 Y b . Both monomials are in Cd0 , where d0 is equal to d
minus the S−degree of gcd(T, Y b ). By Lemma 4, if CdA0 1 6= ∅ then d0 ∈ dN, but
since d0 ≺S d this is not possible. So, if T1 is a mixed monomial and CdA0 1 = ∅,
then CdA0 2 6= ∅. If there exists a pure monomial in CdA0 2 connected to a mixed
monomial in Cd0 , we perform the same process obtaining T2 , Y b2 ∈ Cd00 with T2
a mixed monomial and d00 ≺S d0 . This process can be repeated if there exist a
pure monomial and a mixed monomial in the same connected component. By
degree reasons this cannot be performing indefinitely, an element d(i) ∈ Betti(S)
verifying that ∇d(i) is not connected having a connected component with only
mixed monomials is found. This contradicts Lemma 3.
After examining the structure of the simplicial complexes associated to glued
semigroups, we enunciate a combinatorial characterization by means of the nonconnected simplicial complexes ∇m .
Theorem 6. The semigroup S is the gluing of S1 and S2 if and only if the
following conditions are fulfilled:
1. For all d0 ∈ Betti(S), any connected component of ∇d0 has at least a pure
monomial.
2. There exists a unique d ∈ Betti(S) such that CdA1 6= ∅ 6= CdA2 and the
elements in Cd are pure monomials.
3. For all d0 ∈ Betti(S) \ {d} with CdA0 1 6= ∅ =
6 CdA0 2 , d0 ∈ dN.
Besides, the above d ∈ Betti(S) is the glued degree.
Proof. If S is the gluing of S1 and S2 , the result is obtained from Lemmas 3, 4
and 5.
Conversely, by hypothesis 1 and 3, given d0 ∈ Betti(S)\{d} the set M(IS1 )d0
is constructed from CdA0 1 and M(IS2 )d0 from CdA0 2 as in Construction 1. Analogously, if d ∈ Betti(S), the set M(IS )d is obtained from the union of M(IS1 )d ,
M(IS2 )d and the binomial X γX −Y γY with X γX ∈ CdA1 and Y γY ∈ CdA2 . Finally


G
M(IS1 )m t M(IS2 )m t {X γX − Y γY }
m∈Betti(S)

is a generating set of IS and S is the gluing of S1 and S2 .
6

From Theorem 6 we obtain an equivalent property to Theorem 12 in [5] by
using the language of monomials and binomials.
Corollary 7. Let S be the gluing of S1 and S2 , and X γX − Y γY ∈ IS a glued binomial with S−degree d. The ideal IS is minimally generated by its indispensable
binomials if and only if the following conditions are fulfilled:
• The ideals IS1 and IS2 are minimally generated by their indispensable binomials.
• The element X γX − Y γY is an indispensable binomial of IS .
• For all d0 ∈ Betti(S), the elements of Cd0 are pure monomials.
Proof. Suppose that IS is generated by its indispensable binomials. By [8,
Corollary 6], for all m ∈ Betti(S) the simplicial complex ∇m has only two
vertices. By Contruction 1 ∇d = {{X γX }, {Y γY }} and by Theorem 6 for all
A2
1
d0 ∈ Betti(S) \ {d} the simplicial ∇d0 is equal to ∇A
d0 or ∇d0 . In any case,
γX
γY
X −Y
∈ IS is an indispensable binomial, and IS1 , IS2 are generated by
their indispensable binomials.
Conversely, suppose that IS is not generated by its indispensable binomials.
Then, there exists d0 ∈ Betti(S) \ {d} such that ∇d0 has more than two vertices
in at least two different connected components. By hypothesis, there are not
mixed monomials in ∇d0 and thus:
A2
1
• If ∇d0 is equal to ∇A
d0 (or ∇d0 ), then IS1 (or IS2 ) is not generated by its
indispensable binomials.

• Otherwise, CdA0 1 6= ∅ =
6 CdA0 2 and by Lemma 4, d0 = jd with j ∈ N, therefore
(j−1)γX γY
X
Y
∈ Cd0 which contradicts the hypothesis.
We conclude IS is generated by its indispensable binomials.
The following example taken from [13] illustrates the above results.
Example 8. Let S ⊂ N2 be the semigroup generated by the set
{(13, 0), (5, 8), (2, 11), (0, 13), (4, 4), (6, 6), (7, 7), (9, 9)} .
In this case, Betti(S) is
{(15, 15), (14, 14), (12, 12), (18, 18), (10, 55), (15, 24), (13, 52), (13, 13)}.
Using the appropriated notation for the indeterminates in the polynomial
ring k[x1 , . . . , x4 , y1 , . . . , y4 ] (x1 , x2 , x3 and x4 for the first four generators of
S and y1 , y2 , y3 , y4 for the others), the simplicial complexes associated to the
elements in Betti(S) are those that appear in Table 1. From this table and by
using Theorem 6, the semigroup S is the gluing of h(13, 0), (5, 8), (2, 11), (0, 13)i
and h(4, 4), (6, 6), (7, 7), (9, 9)i and the glued degree is (13, 13). From Corollary
7, the ideal IS is not generated by its indispensable binomials (IS has only four
indispensable binomials).

7

C(15,15) = {y12 y3 , y2 y4 }

C(14,14) = {y12 y2 , y32 }

C(12,12) = {y13 , y22 }

C(10,55) = {x21 x4 , x53 }

∇(15,15)

∇(14,14)

∇(12,12)

∇(10,55)

Ys

Ys

Ys

Xs

C(18,18) = {y12 y2 , y23 ,
y1 y32 , y42 }
∇(18,18)

C(15,24) = {x1 x2 x3 , x32 ,
x3 y1 y4 , x3 y2 y3 }
∇(15,24)

C(13,52) = {x2 x43 , x1 x44 ,
x34 y1 y4 , x34 y2 y3 }
∇(13,52)

C(13,13) = {x1 x4 , y1 y4 ,
y2 y3 }
∇(13,13)

Ys

Xs

Ys

Xs
Ys

Ys

Table 1: Non-connected simplicial complexes associated to Betti(S).

3

Generating glued semigroups

In this section, an algorithm to obtain examples of glued semigroups is given.
Consider A1 = {a1 , . . . , ar } and A2 = {b1 , . . . , bt } two minimal generator sets of
the semigroups T1 and T2 and let Lj = {ρji }i be a basis of ker Tj with j = 1, 2.
Assume that IT1 and IT2 are nontrivial proper ideals of their corresponding
polynomial rings. Consider γX and γY be two nonzero elements in Nr and Nt
respectively2 , and the integer matrix


L1
0
L2  .
(5)
A= 0
γX −γY
Let S be a semigroup such that ker S is the lattice generated by the rows of
the matrix A. This semigroup can be computed by using the Smith Normal
Form (see [11, Chapter 3]). Denote by B1 , B2 two sets of cardinality r and t
respectively satisfying S = hB1 , B2 i and ker(hB1 , B2 i) is generated by the rows
of A.
The following Proposition shows that the semigroup S satisfies one of the
necessary conditions to be a glued semigroup.
Proposition 9. The semigroup S verifies G(hB1 i) ∩ G(hB2 i) = (B1 γX )Z =
(B2 γY )Z with d = B1 γX ∈ hB1 i ∩ hB2 i.
Proof. Use that ker S has a basis as (3) and proceed as in the proof of the
necessary condition of Proposition 2.
Due to B1 ∪ B2 may not be a minimal generating set, this condition does not
assure that S is a glued semigroup. For instance, taking the numerical semigroups T1 = h3, 5i, T2 = h2, 7i and (γX , γY ) = (1, 0, 2, 0), the matrix obtained
2 Note

that γX ∈
/ ker T1 and γY ∈
/ ker T2 because these semigroups are reduced.

8

Xs

from formula (5) is


5
 0
1

−3 0
0
7
0 −2


0
−2 
0

and B1 ∪ B2 = {12, 20, 6, 21} is not a minimal generating set. The following
result solve this issue.
Corollary 10. The semigroup S is a glued semigroup if
r
X

γXi > 1 and

i=1

t
X

γY i > 1.

(6)

i=1

Proof. Suppose that the set of generators B1 ∪ B2 of S is non-minimal, thus
one of its elements is a natural combination of the others. Assume that this
element is the first of B1 ∪ B2 , then there exist λ2 , . . . , λr+t ∈ N such that
B1 (1, −λ2 , . . . , −λr ) = B2 (λr+1 , . . . , λr+t ) ∈ G(hB1 i)∩G(hB2 i). By Proposition
9, there exists λ ∈ Z satisfying B1 (1, −λ2 , . . . , −λr ) = B2 (λr+1 , . . . , λr+t ) =
B1 (λγX ). Since B2 (λr+1 , . . . , λr+t ) ∈ S, λ ≥ 0 and thus
ν = (1 − λγX1 , −λ2 − λγX2 , . . . , −λr − λγXr ) ∈ ker(hB1 i) = ker T1 ,
{z
}
|
≤0

with the following cases:
• If λγX1 = 0, then T1 is not minimally generated which it is not possible
by hypothesis.
• If λγX1 > 1, then 0 > ν ∈ ker T1 , but this is not possible due to T1 is a
reduced semigroup.
• If λγX1 = 1, then λ = γX1 = 1 and
ν = (0, −λ2 − γX2 , . . . , −λr − γXr ) ∈ ker T1 .
{z
}
|
≤0

If λi + γXi 6= 0 for some i = 2, . . . , r, then T1 is not a reduced semigroup.
This implies λi = γXi = 0 for all i = 2, . . . , r.
We have just proved that γX = (1, 0, . . . , 0). In the general case, if S is not
minimally generated it is because either γX or γY are elements in the canonical
bases of Nr or Nt , respectively.
To avoidPthis situation, it is sufficient to take
Pr
t
γX and γY satisfying i=1 γXi > 1 and i=1 γY i > 1.
From the above result we obtain a characterization of glued semigroups: S
is a glued semigroup if and only if ker S has a basis as (3) satisfying Condition
(6).
Example 11. Let T1 = h(−7, 2), (11, 1), (5, 0), (0, 1)i ⊂ Z2 and T2 = h3, 5, 7i ⊂
N be two reduced affine semigroups. We compute their associated lattices
ker T1 = h(1, 2, −3, −4), (2, −1, 5, −3)i and ker T2 = h(−4, 1, 1), (−7, 0, 3)i.

9

If we take γX = (2, 0, 2, 0) and γY = (1, 2, 1), the matrix A is


1 2 −3 −4 0
0
0
 2 −1 5 −3 0
0
0 


 0 0
0
0
−4
1
1 


 0 0
0
0 −7 0
3 
2 0
2
0 −1 −2 −1
and the semigroup S ⊂ Z4 × Z2 is generated by
{(1, −5, 35), (3, 12, −55), (1, 5, −25), (0, 1, 0), (2, 0, 3), (2, 0, 5), (2, 0, 7)}.
|
{z
} |
{z
}
B1

B2

The semigroup S is the gluing of the semigroups hB1 i and hB2 i and ker S is generated by the rows of the above matrix. The ideal IS ⊂ C[x1 , . . . , x4 , y1 , . . . , y3 ]
is generated3 by
{x1 x83 x4 − x32 , x1 x22 − x33 x44 , x21 x53 − x2 x34 , x31 x2 x23 − x77 ,
y1 y3 − y22 , y13 y2 − y32 , y14 − y2 y3 , x21 x23 − y15 y2 },
{z
}
|
glued binomial

then S is really a glued semigroup.

3.1

Generating affine glued semigroups

From Example 11 it be can deduced that the semigroup S is not necessarily
torsion-free. In general, a semigroup T is affine (or equivalently it is torsionfree) if and only if the invariant factors4 of the matrix whose rows are a basis
of ker T are equal to one. Assume zero-columns of the Smith Normal Form of
a matrix are located on its right side. We now show conditions for S being
torsion-free.
Take L1 and L2 the matrices whose rows form a basis of ker T1 and ker T2 ,
respectively and let P1 , P2 , Q1 and Q2 be some matrices with determinant ±1
(i.e. unimodular matrices) such that D1 = P1 L1 Q1 and D2 = P2 L2 Q2 are the
Smith Normal Form of L1 and L2 , respectively. If T1 and T2 are two affine
semigroups, the invariant factors of L1 and L2 are equal to 1. Then

 




D1 0
P1 0 0
L1
0
Q1 0
 0 D2  =  0 P2 0   0
L2 
, (7)
0 Q2
0
γX
γY0
0
0 1
γX −γY
{z
}
|
=:A

0
where γX
= γX Q1 and γY0 = −γY Q2 . Let s1 and s2 be the numbers of zerocolumns of D1 and D2 (s1 , s2 > 0 because T1 and T2 are reduced, see [11,
Theorem 3.14]).

Lemma 12. The semigroup S is an affine semigroup if and only if


0
gcd {γXi
}ri=r−s1 ∪ {γY0 i }ti=t−s2 = 1.
3 See

[14] to compute IS when S has torsion.
invariant factors of a matrix are the diagonal elements of its Smith Normal Form (see
[2, Chapter 2] and [11, Chapter 2]).
4 The

10

Proof. With the conditions fulfilled by T1 , T2 and (γX , γY ), the necessary and
sufficient
condition for the invariant
factors of A to be all equal to one is


0
t
0
r
gcd {γXi }i=r−s1 ∪ {γY i }i=t−s2 = 1.
The following Corollary gives the explicit conditions that γX and γY must
satisfy to construct an affine semigroup.
Corollary 13. The semigroup S is an affine glued semigroup if and only if:
1. T1 and T2 are two affine semigroups.
2. (γX , γY ) ∈ Nr+t .
Pr
Pt
3.
i=1 γXi ,
i=1 γY i > 1.
4. There exist fr−s1 , . . . , fr , gt−s2 , . . . , gt ∈ Z such that
0
0
, . . . , γXr
(fr−s1 , . . . , fr )·(γX(r−s
)+(gt−s2 , . . . , gt )·(γY0 (t−s2 ) , . . . , γY0 t ) = 1.
1)

Proof. It is trivial by the given construction, Corollary 10 and Lemma 12.
Therefore, to obtain an affine glued semigroup it is enough to take two affine
semigroups and any solution (γX , γY ) of the equations of the above corollary.
Example 14. Let T1 and T2 be the semigroups of Example 11. We compute
two elements γX = (a1 , a2 , a3 , a4 ) and γY = (b1 , b2 , b3 ) in order to obtain an
affine semigroup. First of all, we perform a decomposition of the matrix as (7)
by computing the integer Smith Normal Form of L1 and L2 :


1
0
0
0
 0
1
0
0

 0
0
0
0

 0
0
0
0
a1 a1 − 2a2 − a3 −7a1 + 11a2 + 5a3 2a1 + a2 + a4


0
0
0

0
0
0

=
1
0
0


0
1
0
−b1 b1 + 2b2 + 3b3 −3b1 − 5b2 − 7b3





1 0
 2 −1

0 0

0 0
0 0

0 0
0 0
−2 1
7 −4
0 0



0
1 2 −3 −4
 2 −1 5 −3
0


0
 0 0 0 0
0  0 0 0 0
1
a1 a2 a3 a4

0
0
0
0
−4 1
−7 0
−b1 −b2

1 1 −7 2
 0 −2 11 1
0


0 
  0 −1 5 0

1 
0 0 0 1
3 
0 0 0 0
−b3  0 0 0 0
0 0 0 0



0 0 0
0 0 0

0 0 0

0 0 0

1 −1 3 

0 −2 5 
0 −3 7

Second, by Corollary 13, we must find a solution to the system:

a1 + a2 + a3 + a4 > 1



b1 + b2 + b3 > 1
f1 , f2 , g1 ∈ Z



f1 (−7a1 + 11a2 + 5a3 ) + f2 (2a1 + a2 + a4 ) + g1 (−3b1 − 5b2 − 7b3 ) = 1
with a1 , a2 , a3 , a4 , b1 , b2 , b3 ∈ N. Such solution is computed (in less than a second) using FindInstance of Wolfram Mathematica (see [15]):
FindInstance[(−7a1 +11a2 +5a3 )∗f1 +(2a1 +a2 +a4 )∗f2 +(−3b1 −5b2 −7b3 )∗g1 == 1
&& a1+a2+a3+a4 > 1&& b1 +b2 +b3 > 1&& a1 ≥ 0&& a2 ≥ 0&& a3 ≥ 0&& a4 ≥ 0
11

&& b1 ≥ 0&& b2 ≥ 0&& b3 ≥ 0, {a1 , a2 , a3 , a4 , b1 , b2 , b3 , f1 , f2 , g1 }, Integers ]

{{a1 → 0, a2 → 0, a3 → 3, a4 → 0, b1 → 1, b2 → 1, b3 → 0, f1 → 1, f2 → 0, g1 → 0}}
We now take γX = (0, 0, 3, 0) and γY = (1, 1, 0), and construct the matrix


1
2 −3 −4
0
0 0
 2 −1
5 −3
0
0 0 


0
0
0
0
−4
1
1 
A=
.

 0
0
0
0 −7
0 3 
0
0
3
0 −1 −1 0
We have the affine semigroup S ⊂ Z2 which is minimally generated by
{(2, −56), (1, 88), (0, 40), (1, 0), (0, 45), (0, 75), (0, 105)}
|
{z
} |
{z
}
B1

B2

satisfies that ker S is generated by the rows of A and it is the result of gluing
the semigroups hB1 i and hB2 i. The ideal IS is generated by
{x1 x83 x4 − x32 , x1 x22 − x33 x44 , x21 x53 − x2 x34 , x31 x2 x23 − x74 ,
y1 y3 − y22 , y13 y2 − y32 , y14 − y2 y3 , x33 − y1 y2 },
| {z }
glued binomial

therefore, S is a glued semigroup.
All glued semigroups have been computed by using our programm ecuaciones
which is available in [3] (this programm requires Wolfram Mathematica 7 or
above to run).

References
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http://www.uca.es/dpto/C101/pags-personales/
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rings. Manuscripta Math. 3 (1970) 175–193.

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[7] I. Ojeda, A. Vigneron-Tenorio. Simplicial complexes and minimal
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13

