The local loop lemma

Abstract

We prove that an idempotent operation generates a loop from a strongly connected digraph containing directed cycles of all lengths under very mild (local) algebraic assumptions. Using the result, we reprove the existence of a weakest non-trivial idempotent equations, and that a strongly connected digraph with algebraic length 1 compatible with a Taylor term has a loop.Proposition 2.1 (Periodicity lemma). Let a, b be positive integers and x be a word of length at least a + b − gcd(a, b). If x is both a-periodic and b-periodic, it is also gcd(a, b)-periodic.Corollary 2.2. Let x be a word and k ≥ 2 be the shortest period of x. If y is a subword of x such that |y| ≥ 2k − 2. Then k is the shortest period of y.Proof. The word y is k-periodic. To obtain a contradiction, let k ′ ≤ k − 1 be another period of y. Since |y| ≥ k ′ + k − 1, the word y is gcd(k, k ′ )-periodic. Since |y| ≥ k and y is a subword of the k-periodic word x, also x is gcd(k, k ′ )periodic, which contradicts the minimality of k.Corollary 5.4. Let G be a strongly connected digraph with algebraic length 1 compatible with a Taylor operation. Then G has a loop.Proof. Let us denote the Taylor operation as t, and the vertex set of G as A.Since the digraph G is strongly connected and has algebraic length 1, it remains