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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.
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.
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.
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.
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 ...
In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces , such that
An up-down poset Q(a,b) is a generalization of a zigzag poset in which there are a downward orientations for every upward one and b total elements. [5] For instance, Q(2,9) has the elements and relations > > < > > < > >. In this notation, a fence is a partially ordered set of the form Q(1,n).
The first diagram makes clear that the power set is a graded poset.The second diagram has the same graded structure, but by making some edges longer than others, it emphasizes that the 4-dimensional cube is a combinatorial union of two 3-dimensional cubes, and that a tetrahedron (abstract 3-polytope) likewise merges two triangles (abstract 2-polytopes).