Persistence of cyclic left distributive algebras

Abstract

Let A k = A k(∗) denote the left distributive groupoid on {0, 1, …, 2 k − 1} such that a ∗ 1  a + 1 mod 2 k for every a ϵ A k . Let d ≥ 0 and put r = max { i; 2 i divides d}. For a = ∑ a i 2 i ϵ A k , a i ϵ {0, 1}, put ν d ( a) = ∑ a i ν d (2 i ) and v d (2 i ) = 2 ( i + 1) 2 d − 2 i2 d . Then v d : A k → A k2 d is a groupoid homomorphism iff k ≤ 2 2 r + 1 .