Order, Lattices & Set Relations

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Abstractions about ordered and relational structure in sets, including partial orders, lattices, joins, meets, antichains, complements, partitions, and definability. They also cover reflection and combinatorial principles governing large or highly structured collections.

36 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Additively indecomposable ordinal — A nonzero ordinal alpha that cannot be reached or exceeded by adding two smaller ordinals, equivalently an ordinal of the form omega raised to an ordinal power.
• Binary relation — Represent which ordered pairs from two declared sets stand in a relation by selecting a subset of their Cartesian product, enabling converse, composition, closure, and relational properties.
• 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.
• C-minimal theory — A model-theoretic theory whose definable one-variable sets are finite Boolean combinations of cones determined by a ternary C-relation.
• Club principle — A set-theoretic guessing principle asserting a sequence of cofinal subsets that is fully contained in every unbounded set at some indexed stage.
• Complement (set theory) — The set of elements in a declared universe that are not members of a selected set, or the elements of one set left after removing another.
• Complete lattice — A partially ordered set in which every subset, including the empty set, has both a supremum and an infimum.
• Connected relation — A binary relation that compares every pair of distinct elements in at least one direction.
• Continuous function (ordinal theory) — An ordinal-indexed sequence whose value at every limit index equals the supremum of its earlier values, usually considered together with monotonicity in transfinite constructions.
• Diamond principle — A set-theoretic guessing principle asserting a sequence that correctly anticipates every subset of the first uncountable ordinal on a stationary set.
• Duality (order theory) — The order-reversing construction that replaces a partially ordered set by the same elements with every comparison reversed.
• 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.
• Ideal (order theory) — A nonempty directed lower set of a partially ordered set, equivalently in a lattice a lower set closed under finite joins.
• Interval order — A partial order representable by real intervals where one element precedes another exactly when its interval lies completely to the left.
• Join and meet — The least upper bound and greatest lower bound, respectively, of a subset in a partially ordered set when those bounds exist.
• Knaster's condition — A chain condition on a partial order requiring every uncountable subset to contain an uncountable pairwise-compatible or linked subset.
• L(R) — The smallest transitive inner model of ZF containing every ordinal and every real, constructed by iterating definability from the real numbers and used to study determinacy under large-cardinal assumptions.
• 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.
• Mouse (set theory) — A small iterable fine-structural model equipped with extender data, used to approximate large-cardinal universes and build core models.
• Normal function — An ordinal-valued function that is strictly increasing and continuous at limit ordinals, so its value at a limit is the supremum of all earlier values.
• Ordinal collapsing function — A notation-building function that introduces symbols for very large ordinals and systematically collapses them into canonical notations for large countable ordinals.
• Ordinal definable set — A set uniquely definable in some rank-initial universe by a first-order formula using finitely many ordinal parameters.
• Partially ordered set — A set equipped with a reflexive, antisymmetric and transitive binary relation whose elements need not all be comparable.
• Partition of a set — A family of nonempty, pairwise disjoint subsets whose union is the whole underlying set, equivalently the classes of an equivalence relation.
• Pluripolar set — A subset of a complex domain contained in the negative-infinity locus of a nontrivial plurisubharmonic function.
• Reflection principle — A set-theoretic principle asserting that any specified finite collection of truths about the universe of sets already holds in some set-sized rank-initial structure, with stronger variants serving as large-cardinal axioms.
• Reflexive relation — A binary relation on a set that relates every element of the set to itself.
• Soft set — A parameterized family of subsets used to represent attribute-dependent or uncertain classifications.
• 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.
• Square principle — A set-theoretic principle asserting a coherent sequence of short club sets with no single global thread.
• Strong antichain — An antichain whose distinct elements have no common lower bound, or dually no common upper bound, in the ambient poset.
• Strong measure zero set — A set coverable, for every positive length sequence, by intervals whose respective lengths are bounded by that sequence.
• Symmetric relation — A binary relation in which a related pair remains related when its two elements are reversed.
• Theory of conjoint measurement — A representation theory that derives additive numerical scales for interacting attributes from qualitative order relations on their combinations.
• Universal set — A set intended to contain every object admitted by a theory, including every set in the relevant universe and potentially itself.
• XYZ inequality — A correlation inequality constraining relative ordering probabilities for three incomparable elements in a finite partially ordered set.