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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.
If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.. If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions.
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.
The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse. Order-embedding. A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x ≤ y (in P) is equivalent to f(x) ≤ f(y) (in Q). Order isomorphism.
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).
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
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.
A subset F of a partially ordered set (P, ≤) is a filter or dual ideal if the following are satisfied: [3] Nontriviality The set F is non-empty. Downward directed For every x, y ∈ F, there is some z ∈ F such that z ≤ x and z ≤ y. Upward closure For every x ∈ F and p ∈ P, the condition x ≤ p implies p ∈ F.
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