Clones on infinite sets

A universal algebra is a base set X together with functions, or operations, on this set, each of which has a finite number of variables. For example, the base set of the natural numbers with the usual sum as the only operation, the two-element set {0,1} with the logical operation AND, and vector spaces are universal algebras. Whereas other parts of the mathematical field of algebra deal with specific algebraic structures, as for example linear algebra deals with the above-mentioned vector spaces, the field of universal algebra studies mainly the relation between all universal algebras. In this context, two universal algebras are usually considered equivalent if they generate the same term functions from their fundamental functions. If we fix a base set X, then a clone is a class of term equivalent algebras on X. Equivalently, a clone can be defined to be a set of operations on X which contains certain trivial functions, the projections, and which is closed under composition of functions. The clones on X can be ordered by set-theoretical inclusion, and there is a largest clone, the clone of all functions, and a smallest clone, which contains only the projections. In fact, by ordering the clones in such a way, one obtains an algebraic lattice Cl(X). It is the aim of clone theory to understand the structure of Cl(X) for all X. This goal is nothing else than understanding the relation between all universal algebras on X up to term equivalence. For base sets X which have only two elements, this problem has been completely solved. On the contrary, if X has at least three elements, then Cl(X) is already so complex that it seems impossible to fully understand it. Clone theorists thus try to determine certain interesting parts of Cl(X), such as the atoms and the dual atoms, or natural intervals in Cl(X). For finite X, many fundamental questions on Cl(X) were solved in the past years; for example, all dual atoms have been found. On infinite X, not much is known about the clone lattice, although infinite universal algebras are well worth to be investigated: For example, the continuous functions on a topological space are a clone on a usually infinite set. One reason for this is probably that whereas on finite X, the methods needed to investigate Cl(X) are of a purely algebraic nature and thus known to algebraists, on infinite X set-theoretical methods, or methods from mathematical logic, such as infinite combinatorics, Ramsey theory and forcing are required. It is the goal of this project to investigate parts of Cl(X) on infinite X. More specifically, it will focus on clones containing all functions of only one variable, on maximal clones containing all permutations of the base set, on maximal clones closed under conjugation (that is, maximal clones independent of the order of the base set), on a certain partition of the clone lattice, and on a classification of the minimal clones.

A universal algebra is a base set X together with functions, or operations, on this set, each of which has a finite number of variables. For example, the base set of the natural numbers with the usual sum as the only operation, the two-element set {0,1} with the logical operation AND, and vector spaces are universal algebras. Whereas other parts of the mathematical field of algebra deal with specific algebraic structures, as for example linear algebra deals with the above-mentioned vector spaces, the field of universal algebra studies mainly the relation between all universal algebras. In this context, two universal algebras are usually considered equivalent if they generate the same term functions from their fundamental functions. If we fix a base set X, then a clone is a class of term equivalent algebras on X. Equivalently, a clone can be defined to be a set of operations on X which contains certain trivial functions, the projections, and which is closed under composition of functions. The clones on X can be ordered by set-theoretical inclusion, and there is a largest clone, the clone of all functions, and a smallest clone, which contains only the projections. In fact, by ordering the clones in such a way, one obtains an algebraic lattice Cl(X). It is the aim of clone theory to understand the structure of Cl(X) for all X. This goal is nothing else than understanding the relation between all universal algebras on X up to term equivalence. For base sets X which have only two elements, this problem has been completely solved. On the contrary, if X has at least three elements, then Cl(X) is already so complex that it seems impossible to fully understand it. Clone theorists thus try to determine certain interesting parts of Cl(X), such as the atoms and the dual atoms, or natural intervals in Cl(X). For finite X, many fundamental questions on Cl(X) were solved in the past years; for example, all dual atoms have been found. On infinite X, not much is known about the clone lattice, although infinite universal algebras are well worth to be investigated: For example, the continuous functions on a topological space are a clone on a usually infinite set. One reason for this is probably that whereas on finite X, the methods needed to investigate Cl(X) are of a purely algebraic nature and thus known to algebraists, on infinite X set-theoretical methods, or methods from mathematical logic, such as infinite combinatorics, Ramsey theory and forcing are required. It is the goal of this project to investigate parts of Cl(X) on infinite X. More specifically, it will focus on clones containing all functions of only one variable, on maximal clones containing all permutations of the base set, on maximal clones closed under conjugation (that is, maximal clones independent of the order of the base set), on a certain partition of the clone lattice, and on a classification of the minimal clones.