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- Partially_ordered_group abstract "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=0A 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.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.".
- Partially_ordered_group wikiPageExternalLink p071710.htm.
- Partially_ordered_group wikiPageExternalLink Lattice-ordered_group.
- Partially_ordered_group wikiPageID "172048".
- Partially_ordered_group wikiPageRevisionID "599074502".
- Partially_ordered_group hasPhotoCollection Partially_ordered_group.
- Partially_ordered_group subject Category:Order_theory.
- Partially_ordered_group subject Category:Ordered_algebraic_structures.
- Partially_ordered_group subject Category:Ordered_groups.
- Partially_ordered_group type Artifact100021939.
- Partially_ordered_group type Object100002684.
- Partially_ordered_group type OrderedAlgebraicStructures.
- Partially_ordered_group type PhysicalEntity100001930.
- Partially_ordered_group type Structure104341686.
- Partially_ordered_group type Whole100003553.
- Partially_ordered_group type YagoGeoEntity.
- Partially_ordered_group type YagoPermanentlyLocatedEntity.
- Partially_ordered_group comment "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.".
- Partially_ordered_group label "Groupe ordonné".
- Partially_ordered_group label "Grupa uporządkowana".
- Partially_ordered_group label "Grupo ordenado".
- Partially_ordered_group label "Gruppo ordinato".
- Partially_ordered_group label "Partially ordered group".
- Partially_ordered_group label "有序交換群".
- Partially_ordered_group label "順序群".
- Partially_ordered_group sameAs Groupe_ordonné.
- Partially_ordered_group sameAs Gruppo_ordinato.
- Partially_ordered_group sameAs 順序群.
- Partially_ordered_group sameAs Grupa_uporządkowana.
- Partially_ordered_group sameAs Grupo_ordenado.
- Partially_ordered_group sameAs m.01740c.
- Partially_ordered_group sameAs Q2715875.
- Partially_ordered_group sameAs Q2715875.
- Partially_ordered_group sameAs Partially_ordered_group.
- Partially_ordered_group wasDerivedFrom Partially_ordered_group?oldid=599074502.
- Partially_ordered_group isPrimaryTopicOf Partially_ordered_group.