Clones on infinite sets

die die Permutationen von X enthalten. Preface Let X be a set. A fundamental problem of the field of universal algebra is Describe the set of all universal algebras on X. Consider a universal algebra A on X. Many properties of A , such as its congruences, its subalgebras, and its automorphisms, do not depend on the fundamental operations of A , but on its term operations, that is, the operations which are generated from its fundamental operations and the projections by function composition. We therefore call two universal algebras equivalent if and only if they have the same term operations. Up to this equivalence, we can reformulate our problem as follows: Describe the set of all term equivalence classes of universal algebras on X. A clone is a set of term operations of a universal algebra on X. Equivalently, a clone can be defined as a set of finitary operations on X which contains the projections and which is closed under composition. The set of all clones on X thus corresponds to the set of term equivalence classes of universal algebras on X. Ordering this set by set-theoretical inclusion, one obtains a complete algebraic lattice Cl(X). The aim of clone theory is the solution of a certain aspect of the above-mentioned problem, namely Describe Cl(X). This thesis treats instances of the latter question, mainly for infinite X, resulting in several structure theorems on Cl(X). We divide this thesis into an introduction plus three chapters, all of which can be read independently. The chapters correspond to the author's publications [Pin04a], [Pin04b], [Pin0x]. The first chapter deals with clones on a linearly ordered base set X (finite or infinite). Using the linear order, certain natural functions can be defined, the most natural ones being the maximum, the minimum, and the median functions, with their obvious definitions. Whereas it is easily seen that any maximum function of at least two variables v vi generates the maximum functions of all arities, and that the same is true for the minimum functions, it is not clear that the median of, say, three variables generates the median functions of larger arities. Using combinatorial methods, we show that this is indeed the case, that is, all median functions generate the same clone. In the second chapter, we turn to base sets X of infinite regular cardinality. A function is called almost unary iff one of its …