Order Theory

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15 domain-specific abstractions whose origin domain is Order Theory.

• Bounded complete poset — A partially ordered set in which every subset having an upper bound also has a least upper bound, expressing completeness for mutually consistent collections.
• Completely distributive lattice — A complete lattice in which arbitrary meets distribute over arbitrary joins according to the choice-function identity, equivalently satisfying the self-dual complete distributivity law.
• Distributivity (order theory) — A family of order-theoretic laws governing how infima and suprema interact, from finite lattice distributivity to complete and infinite distributive variants.
• Duality (order theory) — The order-reversing construction that replaces a partially ordered set by the same elements with every comparison reversed.
• Frink ideal — A subset I of a partially ordered set such that every common lower bound of the common upper bounds of each finite subset of I also belongs to I.
• Hasse diagram — A drawing of a finite partially ordered set using vertices for elements and upward cover edges while omitting reflexive and transitively implied relations.
• Interval order — A partial order representable by real intervals where one element precedes another exactly when its interval lies completely to the left.
• Law of trichotomy — The order principle that for every two elements exactly one of x<y, x=y or y<x holds, equivalently combining connectedness with asymmetry for a strict order.
• Maximal and minimal elements — Elements of a subset in a preorder that have no strictly greater or strictly lesser comparable member in that subset, without necessarily dominating or being dominated by every member.
• Order type — The isomorphism class of an ordered set under order-preserving bijection, capturing its ordering structure independently of element names.
• Partially ordered set — A set equipped with a reflexive, antisymmetric and transitive binary relation whose elements need not all be comparable.
• Sperner property of a partially ordered set — The property that a graded poset’s largest antichain has the same size as its largest rank level.
• Strong antichain — An antichain whose distinct elements have no common lower bound, or dually no common upper bound, in the ambient poset.
• Well-quasi-ordering — A quasi-order in which every infinite sequence contains an earlier element below a later one, equivalently having neither infinite descending chains nor infinite antichains.
• XYZ inequality — A correlation inequality constraining relative ordering probabilities for three incomparable elements in a finite partially ordered set.