Maximal Clones on Uncountable Sets That Include All Permutations

We first determine the maximal clones on a set X of infinite regular cardinality κ which contain all permutations but not all unary functions, extending a result of Heindorf’s for countably infinite X. If κ is countably infinite or weakly compact, this yields a list of all maximal clones containing the permutations since in that case the maximal clones above the unary functions are known. We then generalize a result of Gavrilov’s to obtain on all infinite X a list of all maximal submonoids of the monoid of unary functions which contain the permutations. 1. Clones and the Results 1.1. The clone lattice. Let X be a set of size |X| = κ. For each natural number n ≥ 1 we denote the set of functions on X of arity n by O(n). We set O = ⋃∞ n=1 O (n) to be the set of all finitary operations on X. A clone is a subset of O which contains the projection maps and which is closed under composition. Since arbitrary intersections of clones are obviously again clones, the set of clones on X forms a complete algebraic lattice Cl(X) which is a subset of the power set of O. The clone lattice is countably infinite if X has exactly two elements, and of size continuum if X is finite and has at least three elements. On infinite X we have |Cl(X)| = 22 κ . The dual atoms of the clone lattice are called maximal clones. On finiteX there exist finitely many maximal clones and an explicit list of those clones has been provided by Rosenberg [6]. Moreover, the clone lattice is dually atomic in that case, that is, every clone is contained in a maximal one. For X infinite the number of maximal clones equals the size of the whole clone lattice ([8], see also [2]), so that it seems impossible to know all of them. It has also been shown [3] that if the continuum hypothesis holds, then not every clone on a countably infinite set is contained in a maximal one. 1.2. Clones containing the bijections. However, even on infinite X the sublattice of Cl(X) of clones containing the set S of all permutations of X 1991 Mathematics Subject Classification. Primary 08A40; secondary 08A05.