Ordered Abelian group

An ordered Abelian group is a mathematical structure . It is an Abelian group , on which there is also an ordering relation compatible with the group structure , which is usually referred to as (one reads less than or equal). This makes it possible to compare the elements of a group according to size . ${\displaystyle \leq }$

Many conceptualizations from the theory of ordered vector spaces can be on Abelian groups transmitted by the one scalar multiplication by - module replaces structure, however, accounts for geometric considerations such Konvexitätsargumente. ${\displaystyle \mathbb {Z} }$

Every ordered Abelian group is torsion-free . Conversely, an Abelian group can be provided with an order so that an ordered Abelian group is obtained if the group is torsion-free.

Ordered Abelian groups are a special case of the more general concept of the ordered group .

definition

An ordered Abelian group is a triple consisting of an Abelian group and a relation , so that the following applies: ${\displaystyle (G,+,\leq )}$ ${\displaystyle (G,+)}$ ${\displaystyle \leq }$

1. It applies to all , that is, is reflexive . ${\displaystyle x\in G}$ ${\displaystyle x\leq x}$ ${\displaystyle \leq }$
2. From and follows for all , that is, is transitive . ${\displaystyle x\leq y}$ ${\displaystyle y\leq z}$ ${\displaystyle x\leq z}$ ${\displaystyle x,y,z\in G}$ ${\displaystyle \leq }$
3. From follows for everyone , that is, is compatible with the group structure. ${\displaystyle x\leq y}$ ${\displaystyle x+z\leq y+z}$ ${\displaystyle x,y,z\in G}$ ${\displaystyle \leq }$

Positive crowd

The set is called the positive set and is a sub - semigroup that contains the neutral element 0. Of course, stands for . ${\displaystyle G^{+}:=\{x\in G;\,x\geq 0\}}$ ${\displaystyle x\geq 0}$ ${\displaystyle 0\leq x}$

Is reversed in an abelian group , a lower half group containing the neutral element, given and defined one by , it is an ordered Abelian group that applies. Accordingly, an ordered Abelian group can also be defined as an Abelian group in which a sub-group is distinguished. Many properties of ordered Abelian groups can be described both by means of the order relation and by means of properties of the subgroup . ${\displaystyle (G,+)}$ ${\displaystyle U}$ ${\displaystyle x\leq y}$ ${\displaystyle y-x\in U}$ ${\displaystyle (G,+,\leq )}$ ${\displaystyle G^{+}=U}$ ${\displaystyle G^{+}}$

Is of finite order , so is it . If all elements of the group have finite order, then a subgroup and the order is nothing more than an equivalence relation . Substantial applications of order theory will therefore only be expected for groups with elements of infinite order; in particular, the groups occurring in the theory are infinite. ${\displaystyle x\in G^{+}}$ ${\displaystyle n}$ ${\displaystyle -x=(n-1)\cdot x\in G^{+}}$ ${\displaystyle G^{+}}$

Positive figures

Let be and two ordered Abelian groups, linkage and order relation are denoted here with the same symbols. ${\displaystyle (G,+,\leq )}$ ${\displaystyle (H,+,\leq )}$

A mapping is called positive or monotonic , if it always follows for all . ${\displaystyle f\colon G\to H}$ ${\displaystyle x\leq y}$ ${\displaystyle f(x)\leq f(y)}$ ${\displaystyle x,y\in G}$

A group homomorphism is positive if and only if . ${\displaystyle f\colon G\to H}$ ${\displaystyle f(G^{+})\subset H^{+}}$

In the category of ordered Abelian groups, the morphisms are the positive group homomorphisms.

More terms

Be an ordered Abelian group. ${\displaystyle (G,+,\leq )}$

Antisymmetric order

The order on is called antisymmetric if it always follows from and . The order is antisymmetric if and only if . ${\displaystyle G}$ ${\displaystyle x\leq y}$ ${\displaystyle y\leq x}$ ${\displaystyle x=y}$ ${\displaystyle G^{+}\cap (-G^{+})=\{0\}}$

Some authors include antisymmetry in the definition and, if there is no antisymmetry, speak of a preorder or an embossed group, for example in. An antisymmetric order is also called a strict order .

Directed order

The order on is called directed if there is always one with and for every two elements . Order is directed precisely when . ${\displaystyle G}$ ${\displaystyle x,y\in G}$ ${\displaystyle z\in G}$ ${\displaystyle x\leq z}$ ${\displaystyle y\leq z}$ ${\displaystyle G=G^{+}-G^{+}}$

Organizational units

An element is an atomic unit, if for every one is having . ${\displaystyle e\in G}$ ${\displaystyle x\in G}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle -n\cdot e\leq x\leq n\cdot e}$

In the example with the natural order, each element is a unit of order. The sequence space , understood as an ordered Abelian group, has no units of order. ${\displaystyle (\mathbb {Z} ,+,\leq )}$ ${\displaystyle \mathbb {N} \setminus \{0\}}$ ${\displaystyle c_{0}}$

Scaled, ordered Abelian groups

A scale in is a subset with the following properties: ${\displaystyle G}$ ${\displaystyle S\subset G^{+}}$

• From follows ${\displaystyle 0\leq x\leq s\in S}$ ${\displaystyle x\in S}$
• ${\displaystyle S}$is directed, i.e. for every two elements there is an with and . ${\displaystyle s_{1},s_{2}\in S}$ ${\displaystyle s\in S}$ ${\displaystyle s_{1}\leq s}$ ${\displaystyle s_{2}\leq s}$
• ${\displaystyle S}$is generating, i.e. each is a finite sum of elements . ${\displaystyle x\in G^{+}}$ ${\displaystyle S}$

The pair is then called a scaled, ordered Abelian group. Such a scale is often defined by a unit of order , then and one writes instead of . In the category of scaled, ordered Abelian groups, the morphisms between and are considered to be positive group homomorphisms for which applies. ${\displaystyle (G,S)}$ ${\displaystyle e}$ ${\displaystyle S=\{x\in G;\,0\leq x\leq e\}}$ ${\displaystyle (G,e)}$ ${\displaystyle (G,S)}$ ${\displaystyle (G,S_{G})}$ ${\displaystyle (H,S_{H})}$ ${\displaystyle f\colon G\to H}$ ${\displaystyle f(S_{G})\subset S_{H}}$

Archimedean order

In analogy to the Archimedean axiom , the order is called on ${\displaystyle G}$

• Archimedean , if it always follows from for all . ${\displaystyle n\cdot x\leq y}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x\leq 0}$
• almost Archimedean , if it always follows for everyone . ${\displaystyle -y\leq n\cdot x\leq y}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x=0}$

If the order is antisymmetric, then Archimedean orders are almost Archimedean.

Unperforated order

If it always follows from for one , the order is called imperforate. ${\displaystyle n\cdot x\geq 0}$ ${\displaystyle n\in \mathbb {N} ,n>0}$ ${\displaystyle x\geq 0}$

Unperforated and antisymmetric groups must be torsion-free , because from for one follows because of the imperforation and , therefore, because of the antisymmetry. ${\displaystyle n\cdot x=0}$ ${\displaystyle n\in \mathbb {N} ,n>0}$ ${\displaystyle x\geq 0}$ ${\displaystyle -x\geq 0}$ ${\displaystyle x=0}$

Archimedean directed groups are imperforate.

Riesz interpolation property

As in the theory of ordered vector spaces, one considers other properties of the order, such as the Riesz interpolation property named after Frigyes Riesz , which means:

• If there are finite subsets with for all , then there is one with for all . (It is sufficient to consider two-element sets .) ${\displaystyle A,B\subset G}$ ${\displaystyle a\leq b}$ ${\displaystyle a\in A,b\in B}$ ${\displaystyle x\in G}$ ${\displaystyle a\leq x\leq b}$ ${\displaystyle a\in A,b\in B}$ ${\displaystyle A,B\subset G}$

An ordered Abelian group with an antisymmetric order is called an association or, more precisely, an association-ordered group if there is a supremum for every two elements . This is an element with and , which is the smallest element with this property, i.e. for each with and follows . One shows that is clearly through and determined. One therefore speaks of the supremum of and and writes for it . Analogously, there are also two elements and the infimum . ${\displaystyle (G,+,\leq )}$ ${\displaystyle x,y\in G}$ ${\displaystyle z\in G}$ ${\displaystyle x\leq z}$ ${\displaystyle y\leq z}$ ${\displaystyle w\in G}$ ${\displaystyle x\leq w}$ ${\displaystyle y\leq w}$ ${\displaystyle z\leq w}$ ${\displaystyle z}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\vee y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\wedge y=-((-x)\vee (-y))}$

Apparently, association-ordered groups have the Riesz interpolation property, the reverse is generally not true. It turns out that association-ordered groups are always distributive associations .

Examples

• The best known example of an ordered Abelian group is the group of integers with the usual order relation. This order is strict and it is . The group homomorphisms on are exactly the figures , where . The positive group homomorphisms are exactly the figures , where . ${\displaystyle (\mathbb {Z} ,+,\leq )}$ ${\displaystyle \mathbb {Z} ^{+}=\mathbb {N} _{0}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle f_{n}:\mathbb {Z} \rightarrow \mathbb {Z} ,x\mapsto nx}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle f_{n}}$ ${\displaystyle n\in \mathbb {N} _{0}}$
• As in the first example are and examples of ordered Abelian groups. ${\displaystyle (\mathbb {Q} ,+,\leq )}$ ${\displaystyle (\mathbb {R} ,+,\leq )}$
• In defining exactly when and . Then there is an ordered Abelian group with . ${\displaystyle (\mathbb {Z} ^{2},+)}$ ${\displaystyle (x_{1},x_{2})\leq (y_{1},y_{2})}$ ${\displaystyle x_{1}\leq y_{1}}$ ${\displaystyle x_{2}\leq y_{2}}$ ${\displaystyle (\mathbb {Z} ^{2},+,\leq )}$ ${\displaystyle (\mathbb {Z} ^{2})^{+}=(\mathbb {N} _{0})^{2}}$
• To define exactly when or and ; this is the so-called lexicographical order . Also is an ordered abelian group that is positive set . ${\displaystyle (\mathbb {Z} ^{2},+)}$ ${\displaystyle (x_{1},x_{2})\preceq (y_{1},y_{2})}$ ${\displaystyle x_{1}<y_{1}}$ ${\displaystyle x_{1}=y_{1}}$ ${\displaystyle x_{2}\leq y_{2}}$ ${\displaystyle (\mathbb {Z} ^{2},+,\preceq )}$ ${\displaystyle \{0\}\times \mathbb {N} _{0}\cup (\mathbb {N} \setminus \{0\})\times \mathbb {Z} }$
• If you consider the sub-subgroup of an Abelian group , the associated order relation is equality. ${\displaystyle \{0\}}$
• If there is a semigroup and the associated Grothendieck group , then the image of in defines a semigroup and thus an order . The group of a ring considered in the K-theory is such a Grothendieck group and therefore naturally an ordered Abelian group. ${\displaystyle H}$ ${\displaystyle G={\mathcal {G}}(H)}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle K_{0}}$
• Every ordered vector space is an ordered Abelian group if one forgets the scalar multiplication and only regards the vector space as an Abelian group.

Applications

• The countable , imperforate, ordered Abelian groups with Riesz's interpolation property are precisely those groups that appear as a group of an AF-C * algebra . ${\displaystyle K_{0}}$
• In valuation theory, for a valuation ring with a quotient field, the factor group of the unit groups is defined with the order if and only if . The positive semigroup is given by the remainder of the elements . ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle K^{*}/A^{*}}$ ${\displaystyle [x]\leq [y]}$ ${\displaystyle yx^{-1}\in A}$ ${\displaystyle A}$

Individual evidence

1. ^ Nicolas Bourbaki : Eléments de Mathématique. Algèbre. Chapitres 1 à 3. Springer, Berlin et al. 2007, ISBN 978-3-540-33849-9 , chapter 2, p. 172.
2. ^ Graham Jameson: Ordered Linear Spaces (= Lecture Notes in Mathematics. Vol. 141, ISSN  0075-8434 ). Springer, Berlin et al. 1970, 1.1.
3. ↑ Kenneth R. Goodearl: Partially Ordered Abelian Groups with Interpolation (= Mathematical Surveys and Monographs. Vol. 20). American Mathematical Society, Providence RI 2010, ISBN 0-8218-1520-2 , chapter 1, Basic Notions.
4. ↑ Kenneth R. Davidson: C * -Algebras by Example (= Fields Institute Monographs. Vol. 6). American Mathematical Society, Providence RI 1996, ISBN 0-8218-0599-1 , IV.3 Dimension Groups.
5. ↑ Kenneth R. Goodearl: Partially Ordered Abelian Groups with Interpolation (= Mathematical Surveys and Monographs. Vol. 20). American Mathematical Society, Providence RI 2010, ISBN 0-8218-1520-2 , set 1.24.
6. ↑ Kenneth R. Goodearl: Partially Ordered Abelian Groups with Interpolation (= Mathematical Surveys and Monographs. Vol. 20). American Mathematical Society, Providence RI 2010, ISBN 0-8218-1520-2 , chapter 2, Interpolation.
7. ^ Graham Jameson: Ordered Linear Spaces (= Lecture Notes in Mathematics. Vol. 141, ISSN  0075-8434 ). Springer, Berlin et al. 1970, sentence 2.2.7.