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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 identity function on any partially ordered set is always an order automorphism.; Negation is an order isomorphism from (,) to (,) (where is the set of real numbers and denotes the usual numerical comparison), since −x ≥ −y if and only if x ≤ y.
The set of all linearly independent subsets of a vector space V, ordered by inclusion. The set of all partial choice functions on a collection of non-empty sets, ordered by restriction. The set of all prime ideals of a ring, ordered by inclusion. The specialization order of any sober space is a dcpo.
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.
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 chain is a totally ordered set or a totally ordered subset of a poset. See also total order. Chain complete. A partially ordered set in which every chain has a least upper bound. Closure operator. A closure operator on the poset P is a function C : P → P that is monotone, idempotent, and satisfies C(x) ≥ x for all x in P. Compact.
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.