On finite algebras with probability limit laws

Abstract

An algebraic system has a probability limit law if the values of terms with independent identically distributed random variables have probability distributions that tend to a certain limit (the limit law) as the number of variables in a term grows. For algebraic systems on finite sets, it is shown that, under some geometric conditions on the set of term value distributions, the existence of a limit law strongly restricts the set of possible operations in the algebraic system. In particular, a system that has a limit law without zero components necessarily consists of quasigroup operations (with arbitrary arity), while the limit law is necessarily uniform. Sufficient conditions are also proved for a system to have a probability limit law, which partly match the necessary ones.