On the permutational representation of general sets of operations by partition lattices
Alfred L. Foster · Transactions of the American Mathematical Society · 1949
An Q partition of U is a logical partition of U (into disjoint subsets or cells) in which the cells remain intact under all the (cellular) monotations induced in the class of cells by the various ground monotations co EE Q. The familiar classical theory of imprimitivity sets is concerned with the study of the Q partitions in the special case in which Q is a group (or merely a set) of permutations('); in this special case, for each given Q-partition the induced cellular monotations are (like the ground permutations W) themselves all necessarily cellular permutations. The present communication shows that this latter theory may be cast in the considerably more general framework in which the set Q may be a quite arbitrary set of monotations-not necessarily permutations. In this general case the various induced cellular monotations, corresponding to a given Q-partition, will no longer necessarily be permutations, as in the classical imprimitivity set case. In this present paper we shall, however, study the structure of just these permutational Q-partitions of U for which all the induced cellular monotations are permutations(2). We shall also be concerned with the structure of the closely related more general class of univoque Q-partitions. Each permutational Q-partition may be thought of as exhibiting the class Q of monotations of U as a set of permutations in the large of U. In this connection it is shown that under specified conditions (which are always satisfied if, for example, U is finite) all Q-permutational partitions are derivable from a unique atomic partition, which is also 2-permutational; that is, any Q-permutational partition has this atomic partition as a refinement. In the