|
In 1941 Emil Post showed that there are only countably many, essentially distinct algebraic structures of size 2. Here we consider two algebras as essentially the same (term equivalent) if they have the same term functions. As an example of term equivalent structures we may consider a Boolean lattice and the corresponding Boolean ring. On a finite set of size 3 or larger, the number of algebraic structures up to term equivalence is continuum already (Yanov, Muchnik, 1959). We show that only countably many of these algebras have a group operation among their term operations. This follows from the observation that the term functions on algebras with group operation are determined by finitely many relations. As an application of these techniques from Universal Algebra we also obtain a new result in Group Theory. More generally we can give finite descriptions for finite Malcev algebras and for the socalled algebras with few subpowers. The latter appeared recently in connection with the Constrained Satisfaction Problem in Computer Science. This is joint work with Erhard Aichinger (JKU Linz) and Ralph McKenzie (Vanderbilt, Nashville).
|
