From Wikipedia, the free encyclopedia
In
abstract algebra , a branch of
mathematics , an affine monoid is a
commutative
monoid that is finitely generated, and is
isomorphic to a submonoid of a
free abelian group
Z
d
,
d
≥
0
{\displaystyle \mathbb {Z} ^{d},d\geq 0}
.
[1] Affine monoids are closely connected to
convex polyhedra , and their associated
algebras are of much use in the algebraic study of these geometric objects.
Affine monoids are finitely generated . This means for a monoid
M
{\displaystyle M}
, there exists
m
1
,
…
,
m
n
∈
M
{\displaystyle m_{1},\dots ,m_{n}\in M}
such that
M
=
m
1
Z
+
+
⋯
+
m
n
Z
+
{\displaystyle M=m_{1}\mathbb {Z_{+}} +\dots +m_{n}\mathbb {Z_{+}} }
.
x
+
y
=
x
+
z
{\displaystyle x+y=x+z}
implies that
y
=
z
{\displaystyle y=z}
for all
x
,
y
,
z
∈
M
{\displaystyle x,y,z\in M}
, where
+
{\displaystyle +}
denotes the
binary operation on the affine monoid
M
{\displaystyle M}
.
Affine monoids are also
torsion free . For an affine monoid
M
{\displaystyle M}
,
n
x
=
n
y
{\displaystyle nx=ny}
implies that
x
=
y
{\displaystyle x=y}
for
n
∈
N
{\displaystyle n\in \mathbb {N} }
, and
x
,
y
∈
M
{\displaystyle x,y\in M}
.
A subset
N
{\displaystyle N}
of a monoid
M
{\displaystyle M}
that is itself a monoid with respect to the operation on
M
{\displaystyle M}
is a submonoid of
M
{\displaystyle M}
.
Every submonoid of
Z
{\displaystyle \mathbb {Z} }
is finitely generated. Hence, every submonoid of
Z
{\displaystyle \mathbb {Z} }
is affine.
The submonoid
{
(
x
,
y
)
∈
Z
×
Z
∣
y
>
0
}
∪
{
(
0
,
0
)
}
{\displaystyle \{(x,y)\in \mathbb {Z} \times \mathbb {Z} \mid y>0\}\cup \{(0,0)\}}
of
Z
×
Z
{\displaystyle \mathbb {Z} \times \mathbb {Z} }
is not finitely generated, and therefore not affine.
The
intersection of two affine monoids is an affine monoid.
If
M
{\displaystyle M}
is an affine monoid, it can be
embedded into a
group . More specifically, there is a unique group
g
p
(
M
)
{\displaystyle gp(M)}
, called the group of differences , in which
M
{\displaystyle M}
can be embedded.
g
p
(
M
)
{\displaystyle gp(M)}
can be viewed as the set of equivalences classes
x
−
y
{\displaystyle x-y}
, where
x
−
y
=
u
−
v
{\displaystyle x-y=u-v}
if and only if
x
+
v
+
z
=
u
+
y
+
z
{\displaystyle x+v+z=u+y+z}
, for
z
∈
M
{\displaystyle z\in M}
, and
(
x
−
y
)
+
(
u
−
v
)
=
(
x
+
u
)
−
(
y
+
v
)
{\displaystyle (x-y)+(u-v)=(x+u)-(y+v)}
defines the addition.
[1]
The rank of an affine monoid
M
{\displaystyle M}
is the
rank of a group of
g
p
(
M
)
{\displaystyle gp(M)}
.
[1]
If an affine monoid
M
{\displaystyle M}
is given as a submonoid of
Z
r
{\displaystyle \mathbb {Z} ^{r}}
, then
g
p
(
M
)
≅
Z
M
{\displaystyle gp(M)\cong \mathbb {Z} M}
, where
Z
M
{\displaystyle \mathbb {Z} M}
is the subgroup of
Z
r
{\displaystyle \mathbb {Z} ^{r}}
.
[1]
If
M
{\displaystyle M}
is an affine monoid, then the monoid
homomorphism
ι
:
M
→
g
p
(
M
)
{\displaystyle \iota :M\to gp(M)}
defined by
ι
(
x
)
=
x
+
0
{\displaystyle \iota (x)=x+0}
satisfies the following
universal property :
for any monoid homomorphism
φ
:
M
→
G
{\displaystyle \varphi :M\to G}
, where
G
{\displaystyle G}
is a group, there is a unique group homomorphism
ψ
:
g
p
(
M
)
→
G
{\displaystyle \psi :gp(M)\to G}
, such that
φ
=
ψ
∘
ι
{\displaystyle \varphi =\psi \circ \iota }
, and since affine monoids are cancellative, it follows that
ι
{\displaystyle \iota }
is an embedding. In other words, every affine monoid can be embedded into a group.
If
M
{\displaystyle M}
is a submonoid of an affine monoid
N
{\displaystyle N}
, then the submonoid
M
^
N
=
{
x
∈
N
∣
m
x
∈
M
,
m
∈
N
}
{\displaystyle {\hat {M}}_{N}=\{x\in N\mid mx\in M,m\in \mathbb {N} \}}
is the integral closure of
M
{\displaystyle M}
in
N
{\displaystyle N}
. If
M
=
M
N
^
{\displaystyle M={\hat {M_{N}}}}
, then
M
{\displaystyle M}
is integrally closed .
The normalization of an affine monoid
M
{\displaystyle M}
is the integral closure of
M
{\displaystyle M}
in
g
p
(
M
)
{\displaystyle gp(M)}
. If the normalization of
M
{\displaystyle M}
, is
M
{\displaystyle M}
itself, then
M
{\displaystyle M}
is a normal affine monoid.
[1]
A monoid
M
{\displaystyle M}
is a normal affine monoid if and only if
R
+
M
{\displaystyle \mathbb {R} _{+}M}
is finitely generated and
M
=
Z
r
∩
R
+
M
{\displaystyle M=\mathbb {Z} ^{r}\cap \mathbb {R} _{+}M}
.
see also:
Group ring
Let
M
{\displaystyle M}
be an affine monoid, and
R
{\displaystyle R}
a commutative
ring . Then one can form the affine monoid ring
R
M
{\displaystyle R[M]}
. This is an
R
{\displaystyle R}
-module with a free basis
M
{\displaystyle M}
, so if
f
∈
R
M
{\displaystyle f\in R[M]}
, then
f
=
∑
i
=
1
n
f
i
x
i
{\displaystyle f=\sum _{i=1}^{n}f_{i}x_{i}}
, where
f
i
∈
R
,
x
i
∈
M
{\displaystyle f_{i}\in R,x_{i}\in M}
, and
n
∈
N
{\displaystyle n\in \mathbb {N} }
.
In other words,
R
M
{\displaystyle R[M]}
is the set of finite sums of elements of
M
{\displaystyle M}
with coefficients in
R
{\displaystyle R}
.
Affine monoids arise naturally from convex polyhedra,
convex cones , and their associated discrete structures.
Let
C
{\displaystyle C}
be a rational
convex cone in
R
n
{\displaystyle \mathbb {R} ^{n}}
, and let
L
{\displaystyle L}
be a
lattice in
Q
n
{\displaystyle \mathbb {Q} ^{n}}
. Then
C
∩
L
{\displaystyle C\cap L}
is an affine monoid.
[1] (Lemma 2.9, Gordan's lemma)
If
M
{\displaystyle M}
is a submonoid of
R
n
{\displaystyle \mathbb {R} ^{n}}
, then
R
+
M
{\displaystyle \mathbb {R} _{+}M}
is a cone if and only if
M
{\displaystyle M}
is an affine monoid.
If
M
{\displaystyle M}
is a submonoid of
R
n
{\displaystyle \mathbb {R} ^{n}}
, and
C
{\displaystyle C}
is a cone generated by the elements of
g
p
(
M
)
{\displaystyle gp(M)}
, then
M
∩
C
{\displaystyle M\cap C}
is an affine monoid.
Let
P
{\displaystyle P}
in
R
n
{\displaystyle \mathbb {R} ^{n}}
be a rational polyhedron,
C
{\displaystyle C}
the
recession cone of
P
{\displaystyle P}
, and
L
{\displaystyle L}
a lattice in
Q
n
{\displaystyle \mathbb {Q} ^{n}}
. Then
P
∩
L
{\displaystyle P\cap L}
is a finitely generated
module over the affine monoid
C
∩
L
{\displaystyle C\cap L}
.
[1] (Theorem 2.12)