PUB - Publikationen an der Universität Bielefeld

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.