Linearly ordered group
Template:Partially ordered group property
Definition
In terms of partially ordered group
A linearly ordered group is a partially ordered group satisfying the following equivalent conditions:
- The underlying partial order is a total order (linear order), i.e., any two elements of the group are comparable.
- For every element of the group, either that element or its inverse is in the positive cone of the group (note that the identity element is the only element such that both that and its inverse are in the positive cone).
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| unperforated partially ordered group | partially ordered group where positive roots of elements in the positive cone are in the positive cone | |||
| lattice-ordered group | partially ordered group whose underlying poset is a lattice | |||
| Riesz group | partially ordered group whose underlying poset satisfies the Riesz interpolation property |