Partially ordered group

In terms of a partial order
A partially ordered group, sometimes called pogroup or po-group, is a set $$G$$ equipped with two structures:


 * A group structure, i.e., a multiplication, identity element, and inverse map
 * A partial order, which we denote by $$\le$$

such that the following compatibility condition is satisfied:

$$a \le b \mbox{ and } c \le d \implies ac \le bd \ \forall \ a,b,c,d \in G$$

Note that this is equivalent to the following two conditions together:

$$a \le b \implies ag \le bg, \qquad a \le b \implies ga \le gb \qquad \ \ \forall \ a,b,g \in G$$

Because most partially ordered groups of interest are abelian groups, we often use additive notation for partially ordered groups even for the non-abelian case. Note that there is nothing in the definition that forces the group to be abelian. In fact, any group with a discrete partial order (where no two distinct elements are comparable) is a partially ordered group.

In terms of a positive cone
A partially ordered group is a group $$G$$ along with a subset $$G^+$$ of $$G$$, called the positive cone of $$G$$, satisfying the following:


 * The identity element of $$G$$ is in $$G^+$$
 * $$G^+$$ is a subsemigroup of $$G$$, i.e., it is closed under the group multiplication
 * $$G^+$$ is a normal subset of $$G$$, i.e., it is a union of conjugacy classes of $$G$$.
 * The only element of $$G^+$$ whose inverse is in $$G^+$$ is the identity element.

Equivalence of definitions

 * Positive cone in terms of partial order: If $$G$$ has partial order $$\le$$ and identity element $$e$$, the positive cone is defined as the set $$\{ x \in G \mid e \le x \}$$.
 * Partial order in terms of positive cone: If $$G$$ has positive cone $$G^+$$, define a \le b \iff $$a^{-1}b \in G^+$$. Note that $$a^{-1}b \in G^+$$ is equivalent to $$ba^{-1} \in G^+$$ because the elements $$a^{-1}b,ba^{-1}$$ are conjugates.