Partially ordered group

This article describes a compatible combination of two structures: group and poset

Definition

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 $\leq$

such that the following compatibility condition is satisfied:

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

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

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

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