# Local loop lemma

## TL;DR

This paper proves that certain algebraic operations guarantee the existence of loops in strongly connected digraphs with specific cycle properties, with implications for algebraic equations and Taylor terms.

## Contribution

It introduces a local loop lemma showing that idempotent operations induce loops in strongly connected digraphs under mild algebraic conditions.

## Key findings

- Idempotent operations generate loops in strongly connected digraphs with all cycle lengths.
- Reproves the existence of the weakest non-trivial idempotent equations.
- Shows that digraphs with algebraic length 1 compatible with a Taylor term contain a loop.

## 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.