Ordered groups

In mathematics, specifically in group theory, an **ordered group** is a group that is equipped with a total order that is compatible with the group operation. This means that not only do the group elements have a way to be compared (one element can be said to be less than, equal to, or greater than another), but the group operation respects this order.

An Archimedean group is an important concept in the field of mathematics, particularly within the context of ordered groups. An ordered group is a group that is equipped with a total order that is compatible with the group operation.

A cyclically ordered group is a mathematical structure that extends the concept of a group by introducing a specific type of total order compatible with the group operation. More formally, a cyclically ordered group is a group \( G \) equipped with a binary relation \( < \) that satisfies certain conditions to ensure that the order is "cyclic.

The Hahn embedding theorem is a result in functional analysis, particularly in the study of ordered vector spaces and topological vector spaces. It is named after the mathematician Hans Hahn. The theorem states that every ordered vector space can be embedded into a space of real-valued functions.

A **linearly ordered group** is a mathematical structure that combines the properties of a group with those of a linear order. More specifically, it is a group \( G \) equipped with a total order \( < \) that is compatible with the group operation.

An ordered field is a field \( F \) equipped with a total order \( \leq \) that is compatible with the field operations. This means that the order satisfies the following properties: 1. **Totality**: For any two elements \( a, b \in F \), one of the following holds: \( a \leq b \) or \( b \leq a \).

An **ordered ring** is a mathematical structure that combines the properties of a ring with a total order. More formally, an ordered ring is defined as a ring \( R \) together with a total order \( \leq \) that satisfies certain compatibility conditions with the ring operations (addition and multiplication).

A **partially ordered group** (POG) is an algebraic structure that combines the concepts of a group and a partial order. Formally, a group \( G \) is equipped with a binary operation (usually denoted as multiplication or addition) and satisfies the group properties—closure, associativity, existence of an identity element, and existence of inverses.

A **Riesz space** (also known as a **vector lattice**) is a specific type of ordered vector space that combines both vector space and lattice structures.