Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.
Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set itself is sometimes called a poset.
and 22 Related for: Partially ordered set information
a synonym of totally orderedset, but refers generally to some sort of totally ordered subsets of a given partiallyorderedset. An extension of a given...
then a ≤ c (transitivity). A set with a partial order on it is called a partiallyorderedset, poset, or just orderedset if the intended meaning is clear...
In abstract algebra, a partiallyordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤"...
(abbreviated inf; plural infima) of a subset S {\displaystyle S} of a partiallyorderedset P {\displaystyle P} is the greatest element in P {\displaystyle...
{\displaystyle S} is again defined dually. In the particular case of a partiallyorderedset, while there can be at most one maximum and at most one minimum...
Directed sets are a generalization of nonempty totally orderedsets. That is, all totally orderedsets are directed sets (contrast partiallyorderedsets, which...
represent a finite partiallyorderedset, in the form of a drawing of its transitive reduction. Concretely, for a partiallyorderedset ( S , ≤ ) {\displaystyle...
specifically order theory, the join of a subset S {\displaystyle S} of a partiallyorderedset P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle...
mathematician Norberto Cuesta Dutari [es]. More generally, if S is a partiallyorderedset, a completion of S means a complete lattice L with an order-embedding...
a partiallyorderedset such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partiallyordered set...
(rankings without ties) and are in turn generalized by (strictly) partiallyorderedsets and preorders. There are several common ways of formalizing weak...
(Y):Y\subseteq X{\text{ and }}Y{\text{ finite}}\right\}.} In the theory of partiallyorderedsets, which are important in theoretical computer science, closure operators...
mathematics, especially in order theory, the cofinality cf(A) of a partiallyorderedset A is the least of the cardinalities of the cofinal subsets of A...
Cartesian product of partiallyorderedsets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. The words in...
In order-theoretic mathematics, a graded partiallyorderedset is said to have the Sperner property (and hence is called a Sperner poset), if no antichain...
chain is a totally orderedset or a totally ordered subset of a poset. See also total order. Chain complete. A partiallyorderedset in which every chain...
1982:168). It states that in any partiallyorderedset, every totally ordered subset is contained in a maximal totally ordered subset. The Hausdorff maximal...
used to refer to at least three similar, but distinct, classes of partiallyorderedsets, characterized by particular completeness properties. Complete partial...