TOPICS

# 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 , where is called the ground set of and is the partial order of .

An element in a partially ordered set is said to be an upper bound for a subset of if for every , we have . Similarly, a lower bound for a subset is an element such that for every , . If there is an upper bound and a lower bound for , then the poset is said to be bounded.

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)

## References

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.

## Referenced on Wolfram|Alpha

Partially Ordered Set

## Cite this as:

Insall, Matt and Weisstein, Eric W. "Partially Ordered Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PartiallyOrderedSet.html