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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.
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice.
If the preordered set (,) also happens to be a partially ordered set (or more generally, if the restriction (,) is a partially ordered set) then is a maximal element of if and only if contains no element strictly greater than ; explicitly, this means that there does not exist any element such that and .
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 geometry: it is equipped with a metric and is flat. A topology: there is a notion of open sets. There are interfaces among these: Its order and, independently, its metric structure induce its topology. Its order and algebraic structure make it into an ordered field.
A partially ordered set is series-parallel if and only if it does not have four elements forming a fence. [ 3 ] Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.
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.