Search results
Results From The WOW.Com Content Network
Conversely, a strict partial order < on may be converted to a non-strict partial order by adjoining all relationships of that form; that is, := < is a non-strict partial order. Thus, if ≤ {\displaystyle \leq } is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by a < b if a ≤ b and a ...
Both of these are special cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation. Moreover, a preorder on a set X {\displaystyle X} can equivalently be defined as an equivalence relation on X {\displaystyle X} , together with a partial order on the set of equivalence class.
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.For most spaces that are considered in practice, namely for all those that satisfy the T 0 separation axiom, this preorder is even a partial order (called the specialization order).
Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities) Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions
This is a general situation in order theory: A given order can be inverted by just exchanging its direction, pictorially flipping the Hasse diagram top-down. This yields the so-called dual, inverse, or opposite order. Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order.
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations . The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves ...
In general, two points x and y are topologically indistinguishable if and only if x ≤ y and y ≤ x, where ≤ is the specialization preorder on X. It follows that a space X is T 0 if and only if the specialization preorder ≤ on X is a partial order. There are numerous partial orders on a finite set. Each defines a unique T 0 topology.
Order, an academic journal on order theory; Dense order, a total order wherein between any unequal pair of elements there is always an intervening element in the order; Glossary of order theory; Lexicographical order, an ordering method on sequences analogous to alphabetical order on words; List of order topics, list of order theory topics