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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 wikiPageLength "1685".
- Ordered_semigroup wikiPageOutDegree "13".
- Ordered_semigroup wikiPageRevisionID "653626280".
- Ordered_semigroup wikiPageWikiLink Category:Ordered_algebraic_structures.
- Ordered_semigroup wikiPageWikiLink Category:Semigroup_theory.
- Ordered_semigroup wikiPageWikiLink Category_theory.
- Ordered_semigroup wikiPageWikiLink Class_(set_theory).
- Ordered_semigroup wikiPageWikiLink Group_(mathematics).
- Ordered_semigroup wikiPageWikiLink Mathematics.
- Ordered_semigroup wikiPageWikiLink Monoid.
- Ordered_semigroup wikiPageWikiLink Monotone_class_theorem.
- Ordered_semigroup wikiPageWikiLink Morphism.
- Ordered_semigroup wikiPageWikiLink Morphisms.
- Ordered_semigroup wikiPageWikiLink Ordered_group.
- Ordered_semigroup wikiPageWikiLink Partial_order.
- Ordered_semigroup wikiPageWikiLink Partially_ordered_group.
- Ordered_semigroup wikiPageWikiLink Partially_ordered_set.
- Ordered_semigroup wikiPageWikiLink Semigroup.
- Ordered_semigroup wikiPageWikiLink Semigroup_homomorphisms.
- Ordered_semigroup wikiPageWikiLinkText "Ordered semigroup".
- Ordered_semigroup wikiPageWikiLinkText "partially ordered algebraic structure".
- Ordered_semigroup hasPhotoCollection Ordered_semigroup.
- Ordered_semigroup wikiPageUsesTemplate Template:Abstract-algebra-stub.
- Ordered_semigroup subject Category:Ordered_algebraic_structures.
- Ordered_semigroup subject Category:Semigroup_theory.
- Ordered_semigroup hypernym Semigroup.
- Ordered_semigroup type Structure.
- 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 wasDerivedFrom Ordered_semigroup?oldid=653626280.
- Ordered_semigroup isPrimaryTopicOf Ordered_semigroup.