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.”
The paper is structured in 5 chapters:
1. Introduction – 2. Definitions and basic properties – 3. Absorption by limit point (Theorem 1 and its proof) – 4. Limit laws and corresponding algebraic properties (4.1. Absorbing quasigroup operations, 4.2. Quasigroup-induced algebras, Theorem 2 and its proof) – 5. Main theorem (Theorem 3 and its proof, Theorem 4, cf. [A. D. Yashunskiĭ, Mosc. Univ. Math. Bull. 74, No. 4, 135–140 (2019; Zbl 1448.60007); translation from Vestn. Mosk. Univ., Ser. I 74, No. 4, 3–9 (2019)]) – Acknowledgment – References (12 references).