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:
- 0 ∈ H
- if a ∈ H and b ∈ H then a+b ∈ H
- if a ∈ H then -x+a+x ∈ H for each x of G
- if a ∈ H and -a ∈ H then a=0
A partially ordered group G with positive cone G+ is said to be unperforated if n · g ∈ G+ for some natural number 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.
A Riesz group is a 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.
Read more about Partially Ordered Group: Examples
Famous quotes containing the words partially, ordered and/or group:
“Before the land rose out of the ocean, and became dry land, chaos reigned; and between high and low water mark, where she is partially disrobed and rising, a sort of chaos reigns still, which only anomalous creatures can inhabit.”
—Henry David Thoreau (1817–1862)
“But one sound always rose above the clamor of busy life and, no matter how much of a tintinnabulation, was never confused and, for a moment lifted everything into an ordered sphere: that of the bells.”
—Johan Huizinga (1872–1945)
“Remember that the peer group is important to young adolescents, and there’s nothing wrong with that. Parents are often just as important, however. Don’t give up on the idea that you can make a difference.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.5 (1985)