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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.
In a slight abuse of terminology, the term is sometimes also used to refer not to such a relation, but to its corresponding partially ordered set. Partially ordered set. A partially ordered set (,), or poset for short, is a set together with a partial order on . Poset. A partially ordered set. Preorder. A preorder is a binary relation that is ...
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:
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 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.
A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear. By checking these properties, one immediately sees that the well-known orders on natural numbers, integers, rational numbers and reals are all orders in the above sense.
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).