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- Ordered_semigroup abstract "In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S.If S is a group and it is ordered as a semigroup, one obtains the notion of ordered group, and similarly if S is a monoid it may be called ordered monoid. The terms posemigroup, pogroup and pomonoid are also in use.Additive semigroup of natural numbers (N,+) and additive group of integers (Z,+) endowed with natural order are examples of a posemigroup and pogroup. On the other hand, (N∪{0},+) with the natural order is a pomonoid. Clearly, every semigroup can be treated as a posemigroup endowed with the trivial (discrete) partial order: '='. The class of all semigroups may therefore be viewed as a subclass of the class of all posemigroups (indeed one may then prefer to denote a posemigroup by a triple (S,•,≤)).One can attribute two types of morphisms (in the sense of category theory) to posemigroups, namely the posemigroup homomorphisms which are 'order preserving' (equivalently monotone) semigroup homomorphisms and the posemigroup order-embeddings that are (besides being semigroup homomorphisms) both order preserving and reflecting.".
- Ordered_semigroup wikiPageID "19132161".
- Ordered_semigroup wikiPageRevisionID "563416399".
- Ordered_semigroup hasPhotoCollection Ordered_semigroup.
- Ordered_semigroup subject Category:Ordered_algebraic_structures.
- Ordered_semigroup subject Category:Semigroup_theory.
- Ordered_semigroup type Artifact100021939.
- Ordered_semigroup type Object100002684.
- Ordered_semigroup type OrderedAlgebraicStructures.
- Ordered_semigroup type PhysicalEntity100001930.
- Ordered_semigroup type Structure104341686.
- Ordered_semigroup type Whole100003553.
- Ordered_semigroup type YagoGeoEntity.
- Ordered_semigroup type YagoPermanentlyLocatedEntity.
- Ordered_semigroup comment "In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S.If S is a group and it is ordered as a semigroup, one obtains the notion of ordered group, and similarly if S is a monoid it may be called ordered monoid.".
- Ordered_semigroup label "Ordered semigroup".
- Ordered_semigroup sameAs m.04jnh79.
- Ordered_semigroup sameAs Q7100714.
- Ordered_semigroup sameAs Q7100714.
- Ordered_semigroup sameAs Ordered_semigroup.
- Ordered_semigroup wasDerivedFrom Ordered_semigroup?oldid=563416399.
- Ordered_semigroup isPrimaryTopicOf Ordered_semigroup.