# Levi's lemma

The uw = x and v = wy case of Levi's lemma

In theoretical computer science and mathematics, especially in the area of combinatorics on words, the Levi lemma states that, for all strings u, v, x and y, if uv = xy, then there exists a string w such that either

uw = x and v = wy (if |u| ≤ |x|)

or

u = xw and wv = y (if |u| ≥ |x|)

That is, there is a string w that is "in the middle", and can be grouped to one side or the other.[1] Levi's lemma is named after Friedrich Wilhelm Levi, who published it in 1944.[2]

## Applications

Levi's lemma can be applied repeatedly in order to solve word equations; in this context it is sometimes called the Nielsen transformation by analogy with the Nielsen transformation for groups. For example, starting with an equation = where x and y are the unknowns, we can transform it (assuming |x| ≥ |y|, so there exists t such that x=yt) to ytα = , thus to = β. This approach results in a graph of substitutions generated by repeatedly applying Levi's lemma. If each unknown appears at most twice, then word equation is called quadratic; in a quadratic word equation the graph obtained by repeatedly applying Levi's lemma is finite, so it is decidable if a quadratic word equation has a solution.[1] (A more general method for solving word equations is Makanin's algorithm.)[1]

## Generalizations

The above is known as the Levi lemma for strings; the lemma can occur in a more general form in graph theory and in monoid theory; for example, there is a more general Levi lemma for traces.[3]

A monoid in which Levi's lemma holds is said to have the equidivisibility property.[4] The free monoid of strings and string concatenation has this property (by Levi's lemma for strings), but by itself equidivisibility is not enough to guarantee that a monoid is free. However an equidivisibile monoid M is free if additionally there exists a homomorphism f from M to the monoid of natural numbers (free monoid on one generator) with the property that the preimage of 0 contains only the identity element of M, i.e. ${\displaystyle f^{-1}(0)=\{1_{M}\}}$. (Note that f simply being a homomorhism does not guarantee this latter property, as there could be multiple elements of M mapped to 0.)[5] A monoid for which such a homorphims exists is also called graded (and the f is called a gradation).[6]