A Note on Maximal Hierarchies

Böcker S, Dress A (2000)
Advances in Mathematics 151(2): 270-282.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Böcker, Sebastian; Dress, AndreasUniBi
Abstract / Bemerkung
Given a (finite or infinite) set X, a collection C subset of or equal to P(X) of subsets of X is called a hierarchy if it satisfies the condition "C-1, C-2 is an element of C double right arrow C-1 boolean AND C-2 is an element of{empty set, C-1, C-2}." 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) "C-1, C-2 is an element of C and #{C is an element of C\C-1 subset of or equal to C subset of or equal to C-2} = 2 double right arrow C-2 - C-1 is an element of C" and (b) "boolean OR C is an element of C as well as boolean AND C is an element of 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 ail one-element sets. (C) 2000 Academic Press.
Stichworte
set systems; finitary set systems; hierarchies; maximal hierarchies; ample set systems
Erscheinungsjahr
2000
Zeitschriftentitel
Advances in Mathematics
Band
151
Ausgabe
2
Seite(n)
270-282
ISSN
0001-8708
Page URI
https://pub.uni-bielefeld.de/record/1619932

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Böcker S, Dress A. A Note on Maximal Hierarchies. Advances in Mathematics. 2000;151(2):270-282.
Böcker, S., & Dress, A. (2000). A Note on Maximal Hierarchies. Advances in Mathematics, 151(2), 270-282. https://doi.org/10.1006/aima.1999.1891
Böcker, S., and Dress, A. (2000). A Note on Maximal Hierarchies. Advances in Mathematics 151, 270-282.
Böcker, S., & Dress, A., 2000. A Note on Maximal Hierarchies. Advances in Mathematics, 151(2), p 270-282.
S. Böcker and A. Dress, “A Note on Maximal Hierarchies”, Advances in Mathematics, vol. 151, 2000, pp. 270-282.
Böcker, S., Dress, A.: A Note on Maximal Hierarchies. Advances in Mathematics. 151, 270-282 (2000).
Böcker, Sebastian, and Dress, Andreas. “A Note on Maximal Hierarchies”. Advances in Mathematics 151.2 (2000): 270-282.

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