Partially Ordered Set
المؤلف:
Dushnik, B. and Miller, E. W
المصدر:
"Partially Ordered Sets." Amer. J. Math. 63,
الجزء والصفحة:
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9-1-2022
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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.
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