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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.
In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω 2 × ω to {0,1} and p < q if p ⊇ q. This poset satisfies the countable chain condition. Forcing with this poset adds ω 2 distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum ...
For example, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small posetal finitely cocomplete *-autonomous category.
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.
Every interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig. 7b), the set {1, 2, 4, 5, 8} is convex, but not an interval. An interval I is bounded if there exist elements , such that I ⊆ [a, b]. Every interval that can be represented in interval notation is ...
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 ...
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).
We can recover the poset S from the nerve NS and the category C from the nerve NC; in this sense simplicial sets generalize posets and categories. Another important class of examples of simplicial sets is given by the singular set SY of a topological space Y. Here SY n consists of all the continuous maps from the standard topological n-simplex ...