On finite algebras with probability limit laws

An algebraic system has a limit probability law if the distributions of the values of terms composed of independent identically distributed random variables tend to a certain limit (limit law) as the number of variables in the term increases. For algebraic systems on finite sets, it is shown that under some additional geometric conditions on the set of distributions of term values, the presence of a limit law restricts significantly the set of possible operations of the algebraic system in question.
In particular, a system with a limit distribution without zero components must consist of quasigroup operations (of arbitrary arity), and the limit distribution itself must be uniform. Sufficient conditions for the existence of a limit probability law in an algebraic system, partially coinciding with the necessary ones, are also proved.