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Definition: Preorder, Partial Order and Poset
Let \(V\) be a set and let \([\preceq"\subseteq V\times V\) be a binary relation. We call \("\preceq "\)
- a preorder quasiorder), if "$\preceq$" is reflexive and transitive,
- a partial order if "$\preceq$" is, in addition, antisymmetric.
A set $V$ with a partial order "$\preceq$" defined on it is called a poset (p_artially _o_rdered _set) and denoted by $(V,\preceq ).$
A note on the notation $(V,\preceq )$
Because $V$ and "$\preceq$" are both sets, a poset $(V,\preceq )$ is nothing else than an ordered pair of two sets. This notation was established by Kuratowski.
Mentioned in:
Chapters: 1
Corollaries: 2
Definitions: 3 4 5 6 7 8 9
Examples: 10 11
Explanations: 12 13 14
Lemmas: 15
Proofs: 16 17 18 19 20 21
Propositions: 22 23 24 25 26
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References
Bibliography
- Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
