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In combinatorial game theory, poset games are mathematical games of strategy, generalizing many well-known games such as Nim and Chomp. [1] In such games, two players start with a poset (a partially ordered set), and take turns choosing one point in the poset, removing it and all points that are greater. The player who is left with no point to ...
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 ...
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.
Thus, an equivalent definition of the dimension of a poset P is "the least cardinality of a realizer of P." It can be shown that any nonempty family R of linear extensions is a realizer of a finite partially ordered set P if and only if, for every critical pair ( x , y ) of P , y < i x for some order < i in R .
In mathematics, a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is a graded poset, or; a poset with the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or; a poset in which all maximal chains have the same ...
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 ...
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.
In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets that have some upper bound also have a least upper bound.Such a partial order can also be called consistently or coherently complete (Visser 2004, p. 182), since any upper bound of a set can be interpreted as some consistent (non-contradictory) piece of information that extends all the ...