Order theory

subset an ordered set that consists of all elements between two given endpoints

pair of mathematical objects; tuple of specific length (tuple length n=2)

function between ordered sets that preserves or reverses the given order

statement equivalent to the axiom of choice, about the existence of a maximal element in a poset with a maximal chain condition

set is called bounded, if it is, in a certain sense, of finite size

set ordered by a transitive, antisymmetric, and reflexive binary relation

branch of mathematics studying ordering relations

method of construction of the real numbers

least (resp. greatest) of majoring (resp. minoring) elements of a partially ordered set (not necessarily existing in all sets)

partial order where all elements can be compared

In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then called a well-ordered set (or woset). In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering.

visual depiction of a partially ordered set

elements of partially ordered sets such that there is not greater and smaller than each other element, respectively (but there can be incomparable elements)

generalization of the way the alphabetical order of words is based on the alphabetical order of their component letters

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

in order theory, a nonempty, downward‐directed, upward‐closed subset of a preordered set

every element of a partially ordered set A which is greater (resp. lower) than every element of a subset B included in A

thumb|x R y defined by xinteger division|//4≤y//4 is a preorder on the [[natural numbers. It corresponds to the equivalence relation x E y defined by x//4=y//4. The set of equivalence classes is partially ordered, and thus can be shown as a Hasse diagram (depicted).]]

elements of partially ordered sets that are greater and smaller than each other element, respectively

preordered set whose every finite subset has an upper bound

relation between pairs of arithmetic functions

In 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. Formally, \operatorname{cf}(A) = \inf \{|B| : B \subseteq A, (\forall x \in A) (\exists y \in B) (x \leq y)\}

In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.

thumb|Hasse diagram of the [[divisors of 210, ordered by the relation is divisor of, with the upper set ↑14 colored dark green. It is a , but not an , as it can be extended to the larger nontrivial filter ↑2, by including also the light green elements. Since ↑2 cannot be extended any further, it is an ultrafilter.]] In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P.

partial order where every two distinct comparable elements have another element between them

mathematical statement

pair of adjoint functors between two preordered sets seen as categories

theorem that the maximum size of an antichain in a finite partial order equals the minimum number of chains into which it can be partitioned

theorem in order and lattice theory

law (every real number is either positive, negative, or zero)

certain topology on totally ordered sets

the proposition, independent of ZFC, that a nonempty unbounded complete dense total order satisfying the countable chain condition is isomorphic to the reals

condition in commutative algebra

group with a compatible partial order

locally finite partially ordered set, used to model discretized Lorentzian spacetime

in discrete mathematics, a pair of positions in a sequence where two elements are out of sorted order

subset of a preordered set that includes all successors of its elements

bijective order-preserving mapping between partially ordered sets

well-quasi-ordering of finite trees

In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, \mathbf{x},\ \mathbf{y} \in \mathbb{R}^n, we say that \mathbf{x} weakly majorizes (or dominates) \mathbf{y} from below, commonly denoted \mathbf{x} \succ_w \mathbf{y}, when \sum_{i=1}^k x_i^{\downarrow} \geq \sum_{i=1}^k y_i^{\downarrow} for all k=1,\,\dots,\,n, where x_i^{\downarrow} denotes the ith largest entry of \mathbf{x}. If \mathbf{x}, \mathbf{y} further satisfy \sum_{i=1}^n x_i = \sum_{i=1}^n y_i, we say that \mathbf{x} majorizes (or dominates) \mathbf{y} , commonly denoted \mathbf{x} \succ \

lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion

totally ordered set that shares certain properties of the real line

term used in mathematical order theory

theorem

Wikimedia list article

ternary relation that is cyclic (if [𝑥,𝑦,𝑧] then [𝑧,𝑥,𝑦]), asymmetric (if [𝑥,𝑦,𝑧] then not [𝑧,𝑦,𝑥]), transitive (if [𝑤,𝑥,𝑦] and [𝑤,𝑦,𝑧] then [𝑤,𝑥,𝑧]) and connected (for distinct 𝑥,𝑦,𝑧, either [𝑥,𝑦,𝑧] or [𝑧,𝑥,𝑦])

mathematical operator

Theorem in order theory

theorem

two related operations on a poset in order theory

order for the terms of a polynomial

in order theory, a nonempty, upward‐directed, downward‐closed subset of a preordered set

generalization of total orderings allowing ties, axiomatized as strict weak orders, total preorders, or ordered partitions

frechet filter

condition in order theory and topology

associative algebra used in combinatorics, a branch of mathematics

for functions between posets

undirected graph linking pairs of comparable elements in a partial order

topology in which the intersection of any family of open sets is open

property of a partially ordered set