The Number of Unary Clones Containing the Permutations on an Infinite Set

We calculate the number of unary clones (submonoids of the full transformation monoid) containing the permutations, on an infinite base set. It turns out that this number is quite large, sometimes as large as the whole clone lattice. 1. Background and the result Fix a set X and consider for all n ≥ 1 the set O(n) of n-ary operations on X. If we take the union O = ⋃ n≥1 O (n) over these sets, we obtain the set of all operations on X of finite arity. A clone is a subset of O which contains all functions of the form π k (x1, . . . , xn) = xk (1 ≤ k ≤ n), called the projections, and which is closed under composition of functions. With the order of set-theoretical inclusion, the clones on X form a complete algebraic lattice Cl(X). We wish to describe this lattice for infinite X, in which case it has cardinality 22 |X| . A clone is called unary iff it contains only essentially unary functions, i.e., functions which depend on only one variable. Unary clones correspond in an obvious way to submonoids of the full transformation monoid O(1) and we shall not distinguish between the two notions in the following. We say that a unary clone C 6= O(1) is precomplete or maximal iff C together with any unary function f ∈ O(1) \ C generate O(1), i.e. iff the smallest clone containing C as well as f is O(1). In [Pin0x], the author determined all precomplete submonoids of the full transformation monoid O(1) which contain the permutations for all infinite X, which was a generalization from the countable ([Gav65]). The number of such clones turned out to be rather small compared with the size of the clone lattice: On infiniteX of cardinality אα there exist 2|α| + 5 precomplete unary clones, so in particular there are only five precomplete unary clones on countably infinite X. Theorem 1. Let X be an infinite set. If X has regular cardinality, then the precomplete submonoids of O(1) which contain the permutations are exactly 1991 Mathematics Subject Classification. Primary 08A40; secondary 08A05.