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

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

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

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

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

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...

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...

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

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

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

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,...

order are comparable. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real...

and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered". The positive integers, the nonnegative integers and the integers...

(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...

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...

In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with 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...

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...

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...

a partial order Partially ordered space – Partially ordered topological space Riesz space – Partially ordered vector space, ordered as a lattice Topological...

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 ...

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

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...

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...

compatible partial order Partially ordered space – Partially ordered topological space Riesz space – Partially ordered vector space, ordered as a lattice, also...

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

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...

called an upward closed set, an upset, or an isotone set in X) of a partially ordered set ( X , ≤ ) {\displaystyle (X,\leq )} is a subset S ⊆ X {\displaystyle...

MR 0875137, Zbl 0624.06021 Fuchs, László (1963), "IV.6. Cyclically ordered groups", Partially ordered algebraic systems, International series of monographs in pure...

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