Partially ordered group

In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order '≤' that is translation-invariant; in other words, '≤' has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b. In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order '≤' that is translation-invariant; in other words, '≤' has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b. An element x of G is called positive element if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and it is called the positive cone of G. So we have a ≤ b if and only if -a+b ∈ G+. By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially ordered group if and only if there exists a subset H (which is G+) of G such that: A partially ordered group G with positive cone G+ is said to be unperforated if n · g ∈ G+ for some positive integer n implies g ∈ G+. Being unperforated means there is no 'gap' in the positive cone G+. If the order on the group is a linear order, then it is said to be a linearly ordered group.If the order on the group is a lattice order, i.e. any two elements have a least upper bound, then it is a lattice-ordered group (shortly l-group, though usually typeset with a script ell: ℓ-group). A Riesz group is an unperforated partially ordered group with a property slightly weaker than being a lattice ordered group. Namely, a Riesz group satisfies the Riesz interpolation property: if x1, x2, y1, y2 are elements of G and xi ≤ yj, then there exists z ∈ G such that xi ≤ z ≤ yj. If G and H are two partially ordered groups, a map from G to H is a morphism of partially ordered groups if it is both a group homomorphism and a monotonic function. The partially ordered groups, together with this notion of morphism, form a category. Partially ordered groups are used in the definition of valuations of fields.