Clones on Groups

Every logical function can be expressed using the operations AND, OR, and NOT. Actually even AND and NOT suffice to build all functions but AND and OR do not. These well known facts are at the very base of logic and computer science. In Boolean logic we consider mappings with 2 truth values. If we allow more than 2 values, we may still ask which functions - if not all - can be constructed from a given set of operations. All functions that can be obtained as superpositions of some given functions form a so-called clone. The investigation of clones has developed from logic and is now firmly established as a part of universal algebra. Often one is interested in the clone that is generated by the operations of some algebraic structure like a ring or a group. This clone consists of all the functions that can be expressed by a term in the language of the algebra. These term functions capture important information on the structure of the algebra like its lattice of subalgebras, congruences, etc. Distinct operations may yield the same clone of term functions, and if they do, then the underlying algebras are said to be term-equivalent. By allowing constants of the algebra in the term representing a function, we obtain the clone of polynomial functions. This is a true generalization of the well-known concept of polynomial functions on commutative rings (e.g., on the real numbers). One aim of this project is it to characterize clones on a finite set that contain the operation of a group. In many situations investigated so far, the full clone of polynomial functions on an expanded group is already determined by its elements that depend only on 2 variables. In general clones are not that well-behaved. An important aspect of our research is it to determine exactly which parameters of a certain class of algebras suffice to describe the term functions or polynomial functions. We hope that advances in this subject will allow us to contribute to the following long-standing problem in universal algebra: Is there a finite set on which there are more than countably many clones that contain a group operation or, more generally, a Mal`cev operation? A negative answer to that question would support the hope that a nice classification of certain algebraic structures up to term equivalence is possible.