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In this poset, 60 is an upper bound (though not a least upper bound) of the subset {,,,}, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is included, this will be the greatest element, since this is a ...
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.
The greatest element of a poset P can be called unit or just 1 (if it exists). Another common term for this element is top. It is the infimum of the empty set and the supremum of P. The dual notion is called zero. Up-set. See upper set. Upper bound. An upper bound of a subset X of a poset P is an element b of P, such that x ≤ b, for all x in X.
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 ...
2. An inductive definition is a definition that specifies how to construct members of a set based on members already known to be in the set, often used for defining recursively defined sequences, functions, and structures. 3. A poset is called inductive if every non-empty ordered subset has an upper bound infinity axiom See Axiom of infinity.
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 ...
Also note that the empty set usually has upper bounds (if the poset is non-empty) and thus a bounded-complete poset has a least element. One may also consider the subsets of a poset which are totally ordered, i.e. the chains. If all chains have a supremum, the order is called chain complete. Again, this concept is rarely needed in the dual form.
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 .