Clones on infinite sets

A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of Cl(X) for infinite X. In the first chapter we prove the combinatorial result that if X is linearly ordered, then the median functions of different arity defined by that order all generate the same clone. The second chapter deals with clones containing the almost unary functions, that is, all functions whose value is determined by one of its variables up to a small set. We show that on X of regular cardinality, the set of such clones is always a countably infinite descending chain. The third chapter generalizes a result due to L. Heindorf from the countable to all uncountable X of regular cardinality, resulting in an explicit list of all clones containing the permutations but not all unary functions of X. Moreover, all maximal submonoids of the full transformation monoid which contain the permutations of X are determined, on all infinite X; this is an extension of a theorem by G. Gavrilov for countable base sets.Comment: 70 pages; Dissertation written at the Vienna University of Technology under the supervision of Martin Goldstern; essentially consists of the author's papers "The clone generated by the median functions", "Clones containing all almost unary functions, "Maximal clones on uncountable sets that include all permutations" which are all available from arXi

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, on some cardinals as large as the whole clone lattice. Moreover we find that, with one exception, even the cardinalities of the intervals between the monoid of all permutations and the maximal submonoids of the full transformation monoid are as large. Whether or not the only exception is of the same cardinality as the other intervals depends on additional axioms of set theory.Comment: 6 page