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A Note on Maximal Hierarchies

✍ Scribed by Sebastian Böcker; Andreas W.M. Dress

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
134 KB
Volume
151
Category
Article
ISSN
0001-8708

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✦ Synopsis

Given a (finite or infinite) set X, a collection C P(X ) of subsets of X is called a hierarchy if it satisfies the condition ` '' In this note, we characterize maximal hierarchies as set systems that contain the empty set, the full set, and all one-element sets, and in addition satisfy either one of the following two requirements: (1) They are ample and finitary hierarchies, i.e., they are hierarchies that satisfy the two conditions (a

as well as C$ # C holds for every non-empty chain C$ contained in C.'' (2) They are minimal among all ample and finitary set systems that contain the empty set, the full set, and all one-element sets.

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