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A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
Expressing the example as a partially ordered set by its Hasse diagram. Using a set of atomic elements, as the set of the playing card suits: B = {♠, ♥, ♦, ♣}; B 1 = {♠, ♥}; B 2 = {♦, ♣}; B 3 = {♣}; C = {B, B 1, B 2, B 3}. The second condition of the formal definition can be checked by combining all pairs:
For example, a common programming idiom in Perl that converts an array to a hash whose values are the sentinel value 1, for use as a set, is: my %elements = map { $_ => 1 } @elements ; Other popular methods include arrays .
In an ordered set, one can define many types of special subsets based on the given order. A simple example are upper sets; i.e. sets that contain all elements that are above them in the order. Formally, the upper closure of a set S in a poset P is given by the set {x in P | there is some y in S with y ≤ x}. A set that is equal to its upper ...
This example also demonstrates why it is not always natural to have a greatest element. The set of all linearly independent subsets of a vector space V, ordered by inclusion. The set of all partial choice functions on a collection of non-empty sets, ordered by restriction. The set of all prime ideals of a ring, ordered by inclusion.
In discrete optimization, a special ordered set (SOS) is an ordered set of variables used as an additional way to specify integrality conditions in an optimization model. . Special order sets are basically a device or tool used in branch and bound methods for branching on sets of variables, rather than individual variables, as in ordinary mixed integer programm
A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.
For, let x be the supremum of a partially ordered set P. If P x has Grundy value zero, then P itself has a nonzero value, by the formula above; in this case, x is a winning move in P. If, on the other hand, P x has a nonzero Grundy value, then there must be a winning move y in P x, such that the Grundy value of (P x) y is zero.