Partially Ordered Set

up:: 020 MOC Mathematics

A set is partially ordered if it has a partial order . That is, this relation is

• Reflexive:
• Transitive:
• Antisymmetric:

Examples

• Given any set , one can induce a partial order in its power set by the subset relation. Note that it is not total in general.

Related

• A set is a Totally Ordered Set if any two elements in can be compared by this order relation.
• An element is said to be maximal if .
  • An example of a maximal element is a union of subsets
• An element is an upper bound of if all elements in are bounded by ─

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References

• Um Curso de Álgebra Linear, Flávio Ulhoa Coelho & Mary Lilian Lourenço. Editora EDUSP.