Ordered group

Summary

An ordered group, or partially ordered group, is a group with a partial order that is preserved by multiplication on both sides. The order is then determined by its positive cone, the set of elements that are at least the identity. Ordered groups include the additive groups of ordered fields, groups of functions with the pointwise order, and lattice-ordered groups such as groups of real functions. They are the common setting of the theory of ordered algebraic structures.