Ordered group

In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a partially-ordered group is a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.

An element x of G is called positive element if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G⁺, and it is called the positive cone of G. So we have a ≤ b if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 -a+b ∈ G⁺.

By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b.

For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially-ordered group if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 there exists a subset H (which is G⁺) of G such that:

• 0 ∈ H
• if a ∈ H and b ∈ H then a+b ∈ H
• if a ∈ H then -x+a+x ∈ H for each x of G
• if a ∈ H and -a ∈ H then a=0

A partially-ordered group G with positive cone G⁺ is said to be unperforated if n · g ∈ G⁺ for some natural number n implies g ∈ G⁺. Being unperforated means there is no "gap" in the positive cone G⁺.

If the order on the group is a linear order, then it is said to be a linearly-ordered group.
If the order on the group is a lattice order, i.e. any two elements have a least upper bound, then it is a lattice-ordered group.

A Riesz group is a unperforated partially-ordered group with a property slightly weaker than being a lattice ordered group. Namely, a Riesz group satisfies the Riesz interpolation property: if x₁, x₂, y₁, y₂ are elements of G and x_i ≤ y_j, then there exists z ∈ G such that x_i ≤ z ≤ y_j.

If G and H are two partially-ordered groups, a map from G to H is a morphism of partially-ordered groups if it is both a group homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

 and a monotonic function

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

. The partially-ordered groups, together with this notion of morphism, form a category

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

.

Partially-ordered groups are used in the definition of valuations of field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

s.

Examples

• An ordered vector space
  Ordered vector space
  In mathematics an ordered vector space or partially ordered vector space is a vector space equipped with a partial order which is compatible with the vector space operations.- Definition:...
  
   is a partially-ordered group
• A Riesz space
  Riesz space
  In mathematics a Riesz space, lattice-ordered vector space or vector lattice is an ordered vector space where the order structure is a lattice....
  
   is a lattice-ordered group
• A typical example of a partially-ordered group is Z
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  ⁿ, where the group operation is componentwise addition, and we write (a₁,...,a_n) ≤ (b₁,...,b_n) if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
   a_i ≤ b_i (in the usual order of integers) for all i=1,...,n.
• More generally, if G is a partially-ordered group and X is some set, then the set of all functions from X to G is again a partially-ordered group: all operations are performed componentwise. Furthermore, every subgroup
  Subgroup
  In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
  
  of G is a partially-ordered group: it inherits the order from G.