transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} consisting of a set X {\displaystyle...

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...

In abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤"...

and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals...

used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial...

mathematical symbols Order theory Partially ordered set Directional symbols Polynomial-time reduction Cooley, Brandon. "Ordered Sets" (PDF) (Lecture note for:...

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...

{\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...

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...

proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily...

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...

a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set...

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set ( X , ≤ ) {\displaystyle...

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...

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...

represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ( S , ≤ ) {\displaystyle...

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...

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...

(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...

(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...

any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset, where "maximal" is with respect to set inclusion...

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....

mathematician Norberto Cuesta Dutari [es]. More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding...

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...

(rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are several common ways of formalizing weak...

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...

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...

specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after...

Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which...

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...