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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.
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.
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:
A k-Sperner poset is a graded poset in which no union of k antichains is larger than the union of the k largest rank levels, [1] or, equivalently, the poset has a maximum k-family consisting of k rank levels. [2] A strict Sperner poset is a graded poset in which all maximum antichains are rank levels. [2]
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
Every partially ordered set can be viewed as a category in a natural way: there is a unique morphism from x to y if and only if x ≤ y. A monotone Galois connection is then nothing but a pair of adjoint functors between two categories that arise from partially ordered sets.
A partial order of dimension 4 (shown as a Hasse diagram) and four total orderings that form a realizer for this partial order. In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements. Given a locally finite poset P we can define its incidence algebra. Elements of the incidence algebra are functions ƒ that assign to each interval [x, y] of P a real number ƒ(x, y).