Partially ordered set

A set equipped with a reflexive, antisymmetric, and transitive relation. Unlike a total order, its relation need not compare every pair of elements.

Binary relation: A relation between elements of sets, defined as a set of ordered pairs. A partial order is a binary relation satisfying three specific axioms.
Hasse diagram: A diagram of a finite partially ordered set that omits reflexive and transitive edges. It displays the order while suppressing comparisons implied by chains.
Topological sorting: A linear ordering of a directed acyclic graph's vertices that places each source before its targets. It extends dependency constraints into a sequence compatible with the partial order.
Preorder: A reflexive and transitive relation that may fail antisymmetry. Dropping antisymmetry allows distinct elements to relate in both directions.
Reflexive relation: A relation in which every element is related to itself. Reflexivity requires every element to be below itself.
Comparable elements: Two elements of a partially ordered set are comparable when one is related to the other. Comparability distinguishes ordered pairs from pairs left unrelated.
Dependency graph: A directed graph whose edges indicate that one item depends on another. Acyclic dependencies induce a partial order on tasks or components.
Strict partial order: An irreflexive and transitive relation that represents strict precedence. It encodes strict comparisons rather than the reflexive relation used in a partial order.
Antisymmetric relation: A relation where elements related in both directions must be equal. Antisymmetry prevents distinct elements from being equivalent in both directions.
Chain (order theory): A subset of a partially ordered set in which every pair of elements is comparable. Chains isolate the parts of the order that behave like total orders.

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