Patchworks

In this note, we want to introduce a concept that as we shall see-exhibits all interesting relationship with hierarchies: We define a collection T sub set of or equal to P(X) of subsets of a set X to be a patchwork if empty se t not equal C subset of or equal to C and boolean AND (C is an element ofC) C not equal empty set implies boolean AND (C is an element ofC) C is an el ement of C and boolean OR (C is an element ofC) C is an element of C. In th is note. we will investigate patchworks C that contain a maximal hierarchy. We will show this holds if and only if (i) the empty subset empty set. all one-element subsets {x} (x is an element of X) of X, and the set X itself belong to C. and (ii) the patchwork is ample, that is, A, B is an element o f C and #{C is an element of C / A subset of or equal to C subset of or equ al to B} = 2 implies B - A is an element of C. (C) Academic Press.