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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.
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 ...
Base.See continuous poset.; Binary relation.A binary relation over two sets is a subset of their Cartesian product.; Boolean algebra.A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.
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).
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 ...