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  2. Preorder - Wikipedia

    en.wikipedia.org/wiki/Preorder

    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.

  3. Partially ordered set - Wikipedia

    en.wikipedia.org/wiki/Partially_ordered_set

    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 ...

  4. Prewellordering - Wikipedia

    en.wikipedia.org/wiki/Prewellordering

    Preorder (Quasiorder) Partial order Total preorder Total order Prewellordering Well-quasi-ordering Well-ordering Lattice Join-semilattice Meet-semilattice Strict partial order Strict weak order Strict total order Symmetric: Antisymmetric: Connected: Well-founded: Has joins: Has meets: Reflexive: Irreflexive: Asymmetric

  5. Binary relation - Wikipedia

    en.wikipedia.org/wiki/Binary_relation

    A partial order is a relation that is reflexive, antisymmetric, and transitive. A strict partial order is a relation that is irreflexive, asymmetric, and transitive. A total order is a relation that is reflexive, antisymmetric, transitive and connected. [37] A strict total order is a relation that is irreflexive, asymmetric, transitive and ...

  6. Linear extension - Wikipedia

    en.wikipedia.org/wiki/Linear_extension

    A preorder is a reflexive and transitive relation. The difference between a preorder and a partial-order is that a preorder allows two different items to be considered "equivalent", that is, both and hold, while a partial-order allows this only when =.

  7. Ideal (order theory) - Wikipedia

    en.wikipedia.org/wiki/Ideal_(order_theory)

    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 of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.

  8. Weak ordering - Wikipedia

    en.wikipedia.org/wiki/Weak_ordering

    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 ...

  9. Specialization (pre)order - Wikipedia

    en.wikipedia.org/wiki/Specialization_(pre)order

    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).