Cyclic Arithmetic and Combinatorial Operations

Despite the ubiquity of the rings ℤ_𝑛 as the finite, cyclic models of integer arithmetic, it is rather less well-known that cyclic models of (positive) arithmetic with exponentiation exist only for cycle length 1, 2, 6, 42, and 1806. We explore finite cyclic models of other arithmetical and combinatorial operations on positive arithmetic: fixed base exponentiation, factorial, and binomial coefficients. In each case we find whether infinitely or finitely many models are possible. The case of fixed base exponentiation is particularly interesting, where for base 𝑏 > 2 we find that the compatible cycle sizes form a multiplicative monoid of positive integers, with infinitely many irreducible elements, and curiously sparse prime factors.