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In Political science and Decision theory, order relations are typically used in the context of an agent's choice, for example the preferences of a voter over several political candidates. x ≺ y means that the voter prefers candidate y over candidate x. x ~ y means the voter is indifferent between candidates x and y.
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:
The set of integers and the set of even integers have the same order type, because the mapping is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there ...
If admits a totally ordered cofinal subset, then we can find a subset that is well-ordered and cofinal in . Any subset of is also well-ordered. Two cofinal subsets of with minimal cardinality (that is, their cardinality is the cofinality of ) need not be order isomorphic (for example if = +, then both + and {+: <} viewed as subsets of have the countable cardinality of the cofinality of but are ...
In a partially ordered set (P,≤) an element c is called compact (or finite) if it satisfies one of the following equivalent conditions: For every directed subset D of P, if D has a supremum sup D and c ≤ sup D then c ≤ d for some element d of D. For every ideal I of P, if I has a supremum sup I and c ≤ sup I then c is an element of I.
Download QR code; Print/export ... a continuous poset is a partially ordered set in which every element is the directed supremum ... p.52, Examples I-1.3 ...
In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.