On an infinite base set X, every ideal of subsets of X can be associated with the clone of those operations on X which map small sets to small sets. We continue earlier investigations on the position of such clones in the clone lattice.
Let X denote an arbitrary second-countable, compact, zero-dimensional space. Our main result says that A' is a graph space, i.e., homeomorphic to the space of all complete subgraphs of a suitable graph. We first characterize graph spaces in terms of the Boolean algebras of their clopen subsets. Then it is proved that each countable Boolean algebra has the… (More)