Math 2.0

Order, lattices, ordered algebraic structures Definition

Ordered semigroup

Builds on SemigroupPartial order

Definition

Let (S,⋅)(S, \cdot) be a semigroup and let ≤\le be a partial order on SS. The triple (S,⋅,≤)(S, \cdot, \le) is an ordered semigroup if for all a,b,c∈Sa, b, c \in S, a≤b  ⟹  ca≤cb and ac≤bc.a \le b \implies ca \le cb \text{ and } ac \le bc.

Remarks

Example: (ℕ,+)(\N, +) with the usual order is an ordered semigroup, since a≤ba \le b gives c+a≤c+bc + a \le c + b and a+c≤b+ca + c \le b + c. Non-example: (ℤ,⋅)(\Z, \cdot) with the usual order is not an ordered semigroup. Indeed 1≤21 \le 2, but (−1)⋅1=−1>−2=(−1)⋅2(-1) \cdot 1 = -1 > -2 = (-1) \cdot 2. An ordered semigroup with an identity element is an ordered monoid. Some authors require the order to be total and call the general case a partially ordered semigroup.

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Binary operation, definition, no evidence levelBinary operationdefinitionElement, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptAssociativity, definition, no evidence levelAssociativitydefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSemigroup, definition, no evidence levelSemigroupdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionOrdered semigroup, definition, no evidence levelOrdered semigroupdefinition
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  1. Binary operation definition
  2. Element definition
  3. Proposition concept
  4. Associativity definition
  5. Set definition
  6. Implication definition
  7. Predicate definition
  8. Semigroup definition
  9. Ordered pair definition
  10. Universal quantifier definition
  11. Cartesian product definition
  12. Subset definition
  13. Binary relation definition
  14. Antisymmetric relation definition
  15. Reflexive relation definition
  16. Transitive relation definition
  17. Preorder definition
  18. Partial order definition
  19. Ordered semigroup definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordered_semigroup
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