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

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

mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant...

with a partial order Partially ordered ring – Ring with a compatible partial order Partially ordered space – Partially ordered topological space Preorder...

In mathematics, a partially ordered space (or pospace) is a topological space X {\displaystyle X} equipped with a closed partial order ≤ {\displaystyle...

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...

additionally the path from v to w contains no other vertex. Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say...

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...

(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. By checking these...

partially ordered ring is a partially ordered ring ( A , ≤ ) {\displaystyle (A,\leq )} where A {\displaystyle A} 's partially ordered additive group is...

locally partially ordered space is studied in Roll (1993); see also Directed topology. A cyclically ordered group is a set with both a group structure...

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

graph is a maximal monotone set. Order theory deals with arbitrary partially ordered sets and preordered sets as a generalization of real numbers. The...

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

monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can...

generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are...

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 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...

Quantale Partially ordered monoid Ordered group Archimedean property Ordered ring Ordered field Artinian ring Noetherian Linearly ordered group Monomial...

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...

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...

generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed)...

{\displaystyle \,\sim .\,} It is readily verified that this yields a partially ordered set. Conversely, from any partial order on a partition of a set S...

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...

types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and...

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...

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 ...

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

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

linear extension may be viewed as an order-preserving bijection from a partially ordered set P {\displaystyle P} to a chain C {\displaystyle C} on the same...