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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.
As Jourdan, Rampon & Jard (1994) observe, the problem of listing all cuts in a partially ordered set can be formulated as a special case of a simpler problem, of listing all maximal antichains in a different partially ordered set. If P is any partially ordered set, let Q be a partial order whose elements contain two copies of P: for each ...
Hasse diagram of the set P of divisors of 60, partially ordered by the relation "x divides y".The red subset = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by P op or P d.This dual order P op is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in P op if and only if y ≤ x holds in P.
Order theory knows many completion procedures to turn posets into posets with additional completeness properties. For example, the ideal completion of a given partial order P is the set of all ideals of P ordered by subset inclusion. This construction yields the free dcpo generated by P.
The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse. Order-embedding. A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x ≤ y (in P) is equivalent to f(x) ≤ f(y) (in Q). Order isomorphism.
An ordered set in which every pair of elements is comparable is called totally ordered. Every subset S of a partially ordered set P can itself be seen as partially ordered by restricting the order relation inherited from P to S. A subset S of a partially ordered set P is called a chain (in P) if it is totally ordered in the inherited order.
The directed preordered set (,) is partially ordered if and only if has exactly one element. All pairs of elements from R {\displaystyle R} are comparable and every element of R {\displaystyle R} is a greatest element (and thus also a maximal element) of ( R , ≤ ) . {\displaystyle (R,\leq ).}