Ad
related to: partial order product order
Search results
Results From The WOW.Com Content Network
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] The product order generalizes ...
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 ...
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order.
A partial order of dimension 4 (shown as a Hasse diagram) and four total orderings that form a realizer for this partial order.. In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
In order theory, the Szpilrajn extension theorem (also called the order-extension principle), proved by Edward Szpilrajn in 1930, [1] states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair ...
There are several ways to take two totally ordered sets and extend to an order on the Cartesian product, though the resulting order may only be partial. Here are three of these possible orders, listed such that each order is stronger than the next: Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a ...
Orders are special binary relations. Suppose that P is a set and that ≤ is a relation on P ('relation on a set' is taken to mean 'relation amongst its inhabitants', i.e. ≤ is a subset of the cartesian product P x P). Then ≤ is a partial order if it is reflexive, antisymmetric, and transitive, that is, if for all a, b and c in P, we have that:
It is NP-complete to determine whether a partial order with multiple sources and sinks can be drawn as a crossing-free Hasse diagram. [8] However, finding a crossing-free Hasse diagram is fixed-parameter tractable when parametrized by the number of articulation points and triconnected components of the transitive reduction of the partial order. [9]