Ordered semigroup (original) (raw)
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. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered". Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".
| Property | Value |
|---|---|
| dbo: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. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered". The positive integers, the nonnegative integers and the integers form respectively a posemigroup, a pomonoid, and a pogroup under addition and the natural ordering. Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=". A morphism or homomorphism of posemigroups is a semigroup homomorphism that preserves the order (equivalently, that is monotonically increasing). (en) |
| dbo:wikiPageID | 19132161 (xsd:integer) |
| dbo:wikiPageLength | 1732 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 978175111 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Monoidal_category dbr:Nonnegative_integer dbc:Semigroup_theory dbr:Mathematics dbr:Monoid dbr:Semigroup dbr:Group_(mathematics) dbc:Ordered_algebraic_structures dbr:Positive_integer dbr:Integer dbr:Partial_order dbr:Semigroup_homomorphism dbr:Ordered_group dbr:Skeletal_category dbr:Monotonically_increasing dbr:Thin_category |
| dbp:wikiPageUsesTemplate | dbt:Math dbt:Short_description dbt:Isbn dbt:Abstract-algebra-stub |
| dcterms:subject | dbc:Semigroup_theory dbc:Ordered_algebraic_structures |
| gold:hypernym | dbr:Semigroup |
| rdf:type | yago:WikicatOrderedAlgebraicStructures yago:Artifact100021939 yago:Object100002684 yago:PhysicalEntity100001930 yago:YagoGeoEntity yago:YagoPermanentlyLocatedEntity yago:Structure104341686 yago:Whole100003553 |
| rdfs: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. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered". Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=". (en) |
| rdfs:label | Ordered semigroup (en) |
| owl:sameAs | freebase:Ordered semigroup yago-res:Ordered semigroup wikidata:Ordered semigroup http://ta.dbpedia.org/resource/ஒழுங்கு_அரைகுலம் https://global.dbpedia.org/id/4sUCG |
| prov:wasDerivedFrom | wikipedia-en:Ordered_semigroup?oldid=978175111&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Ordered_semigroup |
| is dbo:wikiPageRedirects of | dbr:Generalized_continuity dbr:Ordered_monoid dbr:Pomonoid |
| is dbo:wikiPageWikiLink of | dbr:Variety_of_finite_semigroups dbr:Generalized_continuity dbr:Special_classes_of_semigroups dbr:Semigroup_Forum dbr:Ordered_monoid dbr:Pomonoid |
| is foaf:primaryTopic of | wikipedia-en:Ordered_semigroup |