On necessary conditions of probability limit theorems in finite algebras

We consider the conditions for a finite set with a given system of operations (a finite algebra) to be subject to a probability limit theorem, i.e., arbitrary computations with mutually independent random variables have value distributions that tend to a certain limit (limit law) as the number of random variables used in the computation grows. Such behavior may be seen as a generalization of the central limit theorem that holds for sums of continuous random variables. We show that the existence of a limit probability law in a finite algebra has strong implications for its set of operations. In particular, with some geometric exceptions excluded, the existence of a limit law without zero components implies that all operations in the algebra are quasigroup operations and the limit law is uniform.