Let X be a non-empty set. By O(X) we denote the set of all fini-tary operations on X, i.e. all functions X n → X of arbitrary ar-ity n. A clone on X is a subset of O(X) which contains all pro-)) must belong to the clone. In the sequel this superposition will be written as f (g 1 ,. .. , g n). It is well-known that the collection L(X) of all clones on X… (More)
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)