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.

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.