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A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is vacuously true. However, its opposite poset has deviation 1. Let k be an algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in one variable. Since the deviation of this poset is the ...
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.
A power set, partially ordered by inclusion, with rank defined as number of elements, forms a graded poset. In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy the following two properties:
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, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of . The term least element is defined dually , that is, it is an element of S {\displaystyle S} that is smaller than every other element of S . {\displaystyle S.}
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.
A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. (A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset.) In the literature, dcpos sometimes also appear under the label up-complete poset.