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Mathematical set with an ordering
transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} consisting of a set X {\displaystyle
Partially_ordered_set
Group with a compatible partial order
In abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤"
Partially_ordered_group
Order whose elements are all comparable
the partially ordered set is a set of subsets of a given set that is ordered by inclusion, and the term is used for stating properties of the set of the
Total_order
On chains and antichains in partial orders
and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals
Dilworth's_theorem
Subset of incomparable elements
a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a finite partially ordered
Antichain
Partial order with well-ordered predecessors
In set theory, a tree is a partially ordered set ( T , < ) {\displaystyle (T,<)} such that for each t ∈ T {\displaystyle t\in T} , the set { s ∈ T : s
Tree_(set_theory)
Set whose pairs have minima and maxima
subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called
Lattice_(order)
Operators to indicate precedence order
mathematical symbols Order theory Partially ordered set Directional symbols Polynomial-time reduction Cooley, Brandon. "Ordered Sets" (PDF) (Lecture note for:
Ordered_set_operators
Generalized alphabetical order
Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. The words in
Lexicographic_order
Subset of a preorder that contains all larger elements
In mathematics, an upper set S {\displaystyle S} of a partially ordered set X {\displaystyle X} is a subset such that if s is in S and if x in X is larger
Upper_and_lower_sets
Mathematical ordering with upper bounds
Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which
Directed_set
Mathematical proposition equivalent to the axiom of choice
proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily
Zorn's_lemma
In order-theoretic mathematics, a graded partially ordered set is said to have the Sperner property (and hence is called a Sperner poset), if no antichain
Sperner property of a partially ordered set
Sperner_property_of_a_partially_ordered_set
Property of a partially ordered set
is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of
Least-upper-bound_property
Mathematical operator
(Y):Y\subseteq X{\text{ and }}Y{\text{ finite}}\right\}.} In the theory of partially ordered sets, which are important in theoretical computer science, closure operators
Closure_operator
Special subset of a partially ordered set
mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear
Filter_(mathematics)
Branch of mathematics
then a ≤ c (transitivity). A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear
Order_theory
In the mathematical field of order theory, an element a of a partially ordered set with least element 0 is an atom if 0 < a and there is no x such that
Atom_(order_theory)
Size of subsets in order theory
mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A
Cofinality
Visual depiction of a partially ordered set
represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ( S , ≤ ) {\displaystyle
Hasse_diagram
Concept in order theory
specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle
Join_and_meet
Greatest lower bound and least upper bound
(abbreviated inf; pl.: infima) of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the greatest element in P {\displaystyle
Infimum_and_supremum
Ring with a compatible partial order
group of a partially ordered ring is always a partially ordered group. The set of non-negative elements of a partially ordered ring (the set of elements
Partially_ordered_ring
Mathematical ranking of a set
(rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are several common ways of formalizing weak
Weak_ordering
Algebraic object with an ordered structure
ordered space – Partially ordered topological space Preorder field – Algebraic concept in measure theory, also referred to as an algebra of setsPages displaying
Ordered_field
Characterizes the height of any finite partially ordered set
combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains
Mirsky's_theorem
Mathematical property of subsets in order theory
disjoint cofinal subsets of the set of all natural numbers. If a partially ordered set A {\displaystyle A} admits a totally ordered cofinal subset, then we can
Cofinal_(mathematics)
Mathematical phrase
used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial
Complete_partial_order
Method of construction of the real numbers
the rational numbers to any totally ordered set by defining a Dedekind cut as a partition of a totally ordered set into two non-empty parts A and B, such
Dedekind_cut
Concept in mathematics
theory, the greatest element of a subset S {\displaystyle S} of a partially ordered set (poset) is an element of S {\displaystyle S} that is greater than
Greatest element and least element
Greatest_element_and_least_element
Equivalence of partially ordered sets
function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be
Order_isomorphism
inductive set to be a partially ordered set that satisfies the hypothesis of Zorn's lemma when nonempty. In descriptive set theory, an inductive set of real
Inductive_set
Partially ordered topological space
{\displaystyle X} is a partially ordered space. For all x , y ∈ X {\displaystyle x,y\in X} with x ≰ y {\displaystyle x\not \leq y} , there are open sets U , V ⊂ X
Partially_ordered_space
Extreme element of a preorder
{\displaystyle S} is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum
Maximal_and_minimal_elements
Unsolved problem on partial orders
factor of 2/3 or better. Equivalently, in every finite partially ordered set that is not totally ordered, there exists a pair of elements x and y with the
1/3–2/3_conjecture
Order-preserving mathematical function
mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose
Monotonic_function
Mathematical result or axiom on order relations
any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset, where "maximal" is with respect to set inclusion
Hausdorff_maximal_principle
Term in the mathematical area of order theory
mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd
Duality_(order_theory)
Smallest complete lattice containing a partial order
specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after
Dedekind–MacNeille_completion
Collection of mathematical objects of finite size
set of real numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any partially ordered set.
Bounded_set
Property of elements related by inequalities
{\overset {<}{\underset {>}{=}}}}y} is true. A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn
Comparability
Partially ordered set in which all subsets have both a supremum and infimum
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally
Complete_lattice
Mathematical relation inside orderings
mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that
Covering_relation
Topics referred to by the same term
element of an ordered pair (x, y) Partially ordered set Complete partial order Permutation, the act of arranging all the members of a set into some sequence
Order
Partial order with joins
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset
Semilattice
Graph linking pairs of comparable elements in a partial order
not comparable to each other in a partial order. For any strict partially ordered set (S,<), the comparability graph of (S, <) is the graph (S, ⊥) of
Comparability_graph
Reflexive and transitive binary relation
{\displaystyle q} . The partially ordered set ( X / ⇔ , ⇐ ) {\displaystyle \left(X/\Leftrightarrow ,\Leftarrow \right)} is hence also a directed set. See Lindenbaum–Tarski
Preorder
Nonempty, upper-bounded, downward-closed subset
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion
Ideal_(order_theory)
Topics referred to by the same term
Top, written ⊤ or 1, in lattice theory, the greatest element in a partially ordered set Top, down tack, or Tee (symbol), the symbol ⊤ Top quark, the third-generation
Top
Existence of certain infima or suprema of a given poset
properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real
Completeness_(order_theory)
Type of topology in mathematics
weak homotopy equivalent to the order complex of the corresponding partially ordered set. Steiner demonstrated that the equivalence is a contravariant lattice
Alexandrov_topology
Class of mathematical orderings
well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible
Well-order
Type of ordering of a set
covering relation is empty. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real numbers
Dense_order
Particular correspondence between two partially ordered sets
connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical
Galois_connection
Endofunctor on the category of simplicial sets
s(I)} be the set of non-empty finite totally ordered subsets, which itself is partially ordered by inclusion. Every partially ordered set can be considered
Subdivision_(simplicial_set)
Well-quasi-ordering of finite trees
Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. A
Kruskal's_tree_theorem
Mathematical ordering of a partial order
order-preserving bijection from a partially ordered set P {\displaystyle P} to a chain C {\displaystyle C} on the same ground set. A preorder is a reflexive
Linear_extension
Glossary of terms used in branch of mathematics
chain is a totally ordered set or a totally ordered subset of a poset. See also total order. Chain complete. A partially ordered set in which every chain
Glossary_of_order_theory
Collection of mathematical objects
and the negation is the set complement. As for every Boolean algebra, the powerset is also a partially ordered set for set inclusion. It is also a complete
Set_(mathematics)
Vector space with a partial order
In mathematics, an ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with
Ordered_vector_space
Partially ordered set equipped with a rank function
combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy
Graded_poset
Partially ordered set with alternatingly-related elements
In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations:
Fence_(mathematics)
British mathematician
His research interests include random combinatorial structures; partially ordered sets; algorithms; random graphs; discrete mathematics and graph theory
Graham_Brightwell
Non-empty family of sets that is closed under finite unions and subsets
partially ordered set (an ideal on a set X {\displaystyle X} is an ideal on the powerset P ( X ) {\displaystyle {\mathcal {P}}(X)} partially ordered by
Ideal_on_a_set
In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements
Locally_finite_poset
Definition of continuity for functions between posets
In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott)
Scott_continuity
Class equipped with a preorder
concepts generalize respectively those of preordered set, partially ordered set and totally ordered set. However, it is difficult to work with them as in
Preordered_class
order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple
Series-parallel_partial_order
General concept and operation in mathematics
theorem of Galois theory. Given a poset P = (X, ≤) (short for partially ordered set; i.e., a set that has a notion of ordering but in which two elements cannot
Duality_(mathematics)
Maximal proper filter
the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P {\textstyle P} is a certain subset of P , {\displaystyle
Ultrafilter
Topics referred to by the same term
generally refers to the existence of certain suprema or infima of some partially ordered set Complete variety, an algebraic variety that satisfies an analog
Completeness
Certain topology in mathematics
totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the
Order_topology
Method for partitioning partial orders into levels
Coffman–Graham algorithm is an algorithm for arranging the elements of a partially ordered set into a sequence of levels. The algorithm chooses an arrangement
Coffman–Graham_algorithm
Maximal proper filter
Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set P ( X
Ultrafilter_on_a_set
Relation between pairs of arithmetic functions
arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence
Möbius_inversion_formula
Nim and Chomp. In such games, two players start with a poset (a partially ordered set), and take turns choosing one point in the poset, removing it and
Poset_game
Condition in order theory and topology
In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable
Countable_chain_condition
Operation in algebra and mathematics
pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of
Monad_(category_theory)
inequality, extreme value and mathematical optimization. Partially ordered set Preorder Totally ordered set Total preorder Chain Trichotomy Extended real number
List_of_order_theory_topics
Special type of lattice
and z in L: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins
Distributive_lattice
Lattice formed by all integer partitions
Young's lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers
Young's_lattice
Set whose elements all belong to another set
the sense that every partially ordered set ( X , ⪯ ) {\displaystyle (X,\preceq )} is isomorphic to some collection of sets ordered by inclusion. The ordinal
Subset
Graph with tight clique-coloring relation
combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, Kőnig's theorem on matchings, and the Erdős–Szekeres theorem on
Perfect_graph
Type of monotone function
kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute
Order_embedding
Algebraic structure used in logic
bounded lattices (thought of as algebraic structures rather than partially ordered sets) with an operation → satisfying certain identities. The identity
Heyting_algebra
Alternative mathematical ordering
generalization to a locally partially ordered space is studied in Roll (1993); see also Directed topology. A cyclically ordered group is a set with both a group
Cyclic_order
redirect targets Ordered ring Ordered vector space – Vector space with a partial order Partially ordered space – Partially ordered topological space
Ordered topological vector space
Ordered_topological_vector_space
Two collections of integer sequences with no such sequence lying between them
that the partially ordered set of possible growth rates of sequences is not complete. Let ω ω {\displaystyle \omega ^{\omega }} be the set of all sequences
Hausdorff_gap
Node ordering for directed acyclic graphs
a linear extension of a partial order in mathematics. A partially ordered set is just a set of objects together with a definition of the "≤" inequality
Topological_sorting
Smallest transitive relation containing a given binary relation
relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "x was born before y" on the set of all people. Symbolically
Transitive_closure
Type of binary relation
The set N × N of pairs of natural numbers, ordered by (n1, n2) < (m1, m2) if and only if n1 < m1 and n2 < m2. Every class whose elements are sets, with
Well-founded_relation
Relationship between elements of two sets
called the codomain. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x,y), where x is an element of X and y is an element of Y. It
Binary_relation
Abstract strategy game
Frederik Schuh. Chomp is a special case of a poset game where the partially ordered set on which the game is played is a product of total orders with the
Chomp
Banach space with a compatible structure of a lattice
complete Normed vector lattice Riesz space – Partially ordered vector space, ordered as a lattice Lattice (order) – Set whose pairs have minima and maxima Birkhoff
Banach_lattice
Mathematical concept regarding posets in (partial) order theory
In mathematics, an upwards linked set A is a subset of a partially ordered set, P, in which any two of elements A have a common upper bound in P. Similarly
Linked_set
Axiom of set theory
Equivalently, in a partially ordered set, every chain can be extended to a maximal chain. Antichain principle: Every partially ordered set has a maximal antichain
Axiom_of_choice
Isomorphism type of ordered sets
In mathematics, especially in set theory and order theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic
Order_type
Mathematical relation making a non-equal comparison
a ≤ b and b ≤ c, then a ≤ c (transitivity) A set with a partial order is called a partially ordered set. Those are the very basic axioms that every kind
Inequality_(mathematics)
order is also important for identifying suitable topologies on partially ordered sets, as is done in order theory. Consider any topological space X {\displaystyle
Specialization_preorder
Directed graph with no directed cycles
partial orders into DAGs works more generally: for every finite partially ordered set (S, ≤), the graph that has a vertex for every element of S and an
Directed_acyclic_graph
travel, tourism, insurance
PARTIALLY ORDERED-SET
PARTIALLY ORDERED-SET
Boy/Male
Arabic, Australian, Muslim
Ordered; Appointed
Girl/Female
Greek
Murdered Agamemnon.
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from a place in Lancashire, called Ormerod, from the Old Norse personal name Ormr (see Orme 1) or Ormarr (a compound of orm ‘serpent’ + herr ‘army’) + Old English rod ‘clearing’.
Boy/Male
Indian
Ordered, Pasted, Appointed
Boy/Male
English Arthurian Legend
Brave.
Boy/Male
Muslim
Ordered, Pasted, Appointed
Male
English
Old English Arthurian legend name of a Knight of the Round Table who was the illegitimate son and traitor of King Arthur, possibly MORDRED means "sea counsel." He was brother (or half-brother) to Agravain, Gaheris, Gareth, and Gawain, and noted for having crowned himself and married Guinevere while Arthur was waging war on Emperor Lucius of Rome. He was killed by Arthur at the Battle of Camlann.Â
Boy/Male
Hindu, Indian, Telugu
Bordered; Friendly Element
Boy/Male
Hindu
Orderly
Girl/Female
English, Peruvian
Plaster; Powdered
Girl/Female
Indian
Well-arranged, Well-ordered
Boy/Male
Tamil
Mitanshu | மீதாஂஷà¯Â
Bordered, Friendly element
Mitanshu | மீதாஂஷà¯Â
Boy/Male
Indian, Sanskrit
Partially Visible
Girl/Female
Muslim
Well-arranged, Well-ordered
Boy/Male
Tamil
Orderly
Boy/Male
African, Indian, Sanskrit
Clear Spoken Person; Ordered
Male
Arthurian
, a son of Lot; traitor to Arthur.
Boy/Male
Indian
Responsibility; Ordered
Girl/Female
African, Arabic, Muslim
Well-ordered; Well-arranged
Boy/Male
American, British, Christian, English
Brave; Brave Counselor
PARTIALLY ORDERED-SET
PARTIALLY ORDERED-SET
PARTIALLY ORDERED-SET
PARTIALLY ORDERED-SET
PARTIALLY ORDERED-SET
PARTIALLY ORDERED-SET
PARTIALLY ORDERED-SET
travel, tourism, insurance