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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.
Any function from a discrete order, i.e. from a set ordered by the identity order "=", is also monotone. Mapping each natural number to the corresponding real number gives an example for an order embedding. The set complement on a powerset is an example of an antitone function.
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 set equipped with a total order is a totally ordered set; [5] the terms simply ordered set, [2] linearly ordered set, [3] [5] toset [6] and loset [7] [8] are also used. The term chain is sometimes defined as a synonym of totally ordered set , [ 5 ] but generally refers to a totally ordered subset of a given partially ordered set.
The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in "discrete" steps and ...
The term complete partial order, abbreviated cpo, has several possible meanings depending on context. A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. (A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset.)
Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which all subsets
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.. The size of the largest antichain in a partially ordered set is known as its width.