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

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

defined as a synonym of totally ordered set, but generally refers to a totally ordered subset of a given partially ordered set. An extension of a given partial...

In set theory, a tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <. Frequently...

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...

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

(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...

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

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

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...

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...

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...

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...

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...

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

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...

(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...

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...

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....

mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear...

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...

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...

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

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

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...

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...

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...

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

{\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...