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A nontrivial poset satisfying the descending chain condition is said to have deviation 0. 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 ...
denote the poset formed by removing x from P. A poset game on P, played between two players conventionally named Alice and Bob, is as follows: Alice chooses a point x ∈ P; thus replacing P with P x, and then passes the turn to Bob who plays on P x, and passes the turn to Alice. A player loses if it is their turn and there are no points to choose.
If used, it requires further definition. Down-set. See lower set. Dual. For a poset (P, ≤), the dual order P d = (P, ≥) is defined by setting x ≥ y if and only if y ≤ x. The dual order of P is sometimes denoted by P op, and is also called opposite or converse order. Any order theoretic notion induces a dual notion, defined by applying ...
As another example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element. This partially ordered set does not even have any maximal elements, since any g divides for instance 2 g , which is distinct from it, so g is not maximal.
Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value. [1] [2] Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram.
In mathematics, a differential poset is a partially ordered set (or poset for short) satisfying certain local properties. (The formal definition is given below.) 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.
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 . An ideal is principal if and only if it is compact in the ideal completion, so the original poset can be recovered as the sub-poset consisting of compact elements.
A partially ordered set (poset) consists of a set of elements together with a binary relation x ≤ y on pairs of elements that is reflexive (x ≤ x for every x), transitive (if x ≤ y and y ≤ z then x ≤ z), and antisymmetric (if both x ≤ y and y ≤ x hold, then x = y).