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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.
Given a set and a partial order relation, typically the non-strict partial order , we may uniquely extend our notation to define four partial order relations , <,, and >, where is a non-strict partial order relation on , < is the associated strict partial order relation on (the irreflexive kernel of ), is the dual of , and > is the dual of <.
The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order. [3] The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. [7]
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 concepts are equivalent although in some circumstances one can be more convenient to work with than the other.
1 Partial order or preorder? 2 comments. 2 Ordering relation vs ordered set. 3 comments. Toggle the table of contents. Talk: Product order. Add languages.
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