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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.
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
The disjoint union of two posets is another typical example of order construction, where the order is just the (disjoint) union of the original orders. Every partial order ≤ gives rise to a so-called strict order <, by defining a < b if a ≤ b and not b ≤ a. This transformation can be inverted by setting a ≤ b if a < b or a = b. The two ...
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 ...
A partial order that includes four elements forming two two-element chains would lead to a relation (, <) that violates the second semiorder axiom, and a partial order that includes four elements forming a three-element chain and an unrelated item would violate the third semiorder axiom (cf. pictures in section #Axiomatics).
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 =.
Ganter & Kuznetsov (1998) describe an incremental algorithm, in which the input partial order is built up by adding one element at a time; at each step, the completion of the smaller partial order is expanded to form the completion of the larger partial order. In their method, the completion is represented by an explicit list of cuts.
Partially order P by: A ≤ B if there exists a strictly increasing f : A → B. Then the subset of non-atomic partial orders forms a filter. Likewise, if I is the set of injective modules over some given commutative ring, of limited cardinality, modulo isomorphism, then a partial order on I is: A ≤ B if there exists an injective linear map f ...