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Standard examples of posets arising in mathematics include: The real numbers , or in general any totally ordered set, ordered by the standard less-than-or-equal relation ≤, is a partial order. On the real numbers R {\displaystyle \mathbb {R} } , the usual less than relation < is a strict partial order.
Some examples of graded posets (with the rank function in parentheses) are: Natural numbers N with their usual order (rank: the number itself), or some interval [0, N] of this poset; N n with the product order (sum of the components), or a subposet of it that is a product of intervals
This family of posets was introduced by Stanley (1988) as a generalization of Young's lattice (the poset of integer partitions ordered by inclusion), many of whose combinatorial properties are shared by all differential posets. In addition to Young's lattice, the other most significant example of a differential poset is the Young–Fibonacci ...
For example, when talking about posets with least element, it may seem reasonable to consider only monotonic functions that preserve this element, i.e. which map least elements to least elements. If binary infima ∧ exist, then a reasonable property might be to require that f ( x ∧ y ) = f ( x ) ∧ f ( y ), for all x and y .
Then, inductively, a poset is said to have deviation at most α (for an ordinal α) if for every descending chain of elements a 0 > a 1 >... all but a finite number of the posets of elements between a n and a n+1 have deviation less than α. The deviation (if it exists) is the minimum value of α for which this is true.
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. A mapping f: P → Q between two posets P and Q is called an order isomorphism, if it is bijective and both f and f −1 are monotone functions.
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.
Examples [ edit ] The face lattice of a convex polytope , consisting of its faces, together with the smallest element, the empty face, and the largest element, the polytope itself, is an Eulerian lattice.