Partially ordered set

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.

Transitive binary relations v t e Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation ✗ ✗ ✗ ✗ ✗ ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ Partial order ✗ ✗ ✗ ✗ ✗ ✗ ✗ Total preorder ✗ ✗ ✗ ✗ ✗ ✗ ✗ Total order ✗ ✗ ✗ ✗ ✗ ✗ Prewellordering ✗ ✗ ✗ ✗ ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ ✗ ✗ ✗ ✗ Well-ordering ✗ ✗ ✗ ✗ ✗ Lattice ✗ ✗ ✗ ✗ ✗ Join-semilattice ✗ ✗ ✗ ✗ ✗ ✗ Meet-semilattice ✗ ✗ ✗ ✗ ✗ ✗ Strict partial order ✗ ✗ ✗ ✗ ✗ ✗ Strict weak order ✗ ✗ ✗ ✗ ✗ ✗ Strict total order ✗ ✗ ✗ ✗ ✗…

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• Katherine Huang
• Marko Dimitrić

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