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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.
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).
Base.See continuous poset.; Binary relation.A binary relation over two sets is a subset of their Cartesian product.; Boolean algebra.A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.
In order-theoretic mathematics, the deviation of a poset is an ordinal number measuring the complexity of a poset. A poset is also known as a partially ordered set. The deviation of a poset is used to define the Krull dimension of a module over a ring as the deviation of its poset of submodules.
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: The rank function is compatible with the ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and
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, specifically category theory, a posetal category, or thin category, [1] is a category whose homsets each contain at most one morphism. [2] As such, a posetal category amounts to a preordered class (or a preordered set , if its objects form a set ).
The subsets of a set form a partially ordered set (poset) for inclusion. Closure operators allow generalizing the concept of closure to any partially ordered set. Given a poset S whose partial order is denoted with ≤ , a closure operator on S is a function C : S → S {\displaystyle C:S\to S} that is