Partially Ordered Set

A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair P=(X,<=), where X is called the ground set of P and <= is the partial order of P.

An element u in a partially ordered set (X,<=) is said to be an upper bound for a subset S of X if for every s in S, we have s<=u. Similarly, a lower bound for a subset S is an element l such that for every s in S, l<=s. If there is an upper bound and a lower bound for X, then the poset (X,<=) is said to be bounded.

See also

Circle Order, Cover Relation, Dominance, Ground Set, Hasse Diagram, Interval Order, Isomorphic Posets, Lattice-Ordered Set, Order Isomorphic, Partial Order, Partially Ordered Multiset, Poset Dimension, Realizer, Relation

Portions of this entry contributed by Matt Insall (author's link)

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Dushnik, B. and Miller, E. W. "Partially Ordered Sets." Amer. J. Math. 63, 600-610, 1941.Fishburn, P. C. Interval Orders and Interval Sets: A Study of Partially Ordered Sets. New York: Wiley, 1985.Skiena, S. "Partial Orders." §5.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 203-209, 1990.Trotter, W. T. Combinatorics and Partially Ordered Sets: Dimension Theory. Baltimore, MD: Johns Hopkins University Press, 1992.

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Partially Ordered Set

Cite this as:

Insall, Matt and Weisstein, Eric W. "Partially Ordered Set." From MathWorld--A Wolfram Web Resource.

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