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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.
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 ...
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
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.
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 .
The best-known example is the existence of all suprema, which is in fact equivalent to the existence of all infima. Indeed, for any subset X of a poset, one can consider its set of lower bounds B . The supremum of B is then equal to the infimum of X : since each element of X is an upper bound of B , sup B is smaller than all elements of X , i.e ...
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.