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|)
For the statement in real analysis, see Beppo Levi's lemma.
The uw = x and v = wy case of Levi's lemma

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. Levi's lemma is named after Friedrich Wilhelm Levi, who published it in 1944.[1]

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 xα = yβ where x and y are the unknowns, we can transform it (assuming |x| ≥ |y|, so there exists t such that x=yt) to ytα = yβ, thus to tα = β. This approach results in a graph of substitutions generated by repeatedly applying Levi's lemma. If each unknown appears at most twice, then a 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.[2] A more general method for solving word equations is Makanin's algorithm.[3][4]

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 originally due to Christine Duboc.[5] Several proofs of Levi's Lemma for traces can be found in The Book of Traces.[6]

A monoid in which Levi's lemma holds is said to have the equidivisibility property.[7] 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 equidivisible 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. . (Note that f simply being a homomorphism does not guarantee this latter property, as there could be multiple elements of M mapped to 0.)[8] A monoid for which such a homomorphism exists is also called graded (and the f is called a gradation).[9]

See also

Notes
  1. Levi, F. W. (1944), "On semigroups", Bulletin of the Calcutta Mathematical Society, 36: 141–146, MR 0011694, Zbl 0061.02405.
  2. Matiyasevich, Yu. V. (1968), "A connection between systems of word and length equations and Hilbert's tenth problem", Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 8: 132–144.
  3. Makanin, G. S. (1977), English transl. in Math. USSR Sbornik 32 (1977), "The problem of solvability of equations in a free semigroup", Mat. Sbornik, 103 (2): 147–236, Bibcode:1977SbMat..32..129M, doi:10.1070/SM1977v032n02ABEH002376
  4. M. Lothaire (2002). "12". Algebraic Combinatorics on Words. Cambridge University Press. ISBN 0-521-81220-8.
  5. Duboc, Chr. (1986), "On some equations in free partially commutative monoids", Theoretical Computer Science, 46: 159–174, doi:10.1016/0304-3975(86)90028-9
  6. Volker Diekert; Grzegorz Rozenberg, eds. (1995). The Book of Traces. World Scientific. pp. 1–576. ISBN 981-02-2058-8.
  7. Aldo de Luca; Stefano Varricchio (1999). Finiteness and Regularity in Semigroups and Formal Languages. Springer Berlin Heidelberg. p. 2. ISBN 978-3-642-64150-3.
  8. M. Lothaire (1997). Combinatorics on Words. Cambridge University Press. p. 13. ISBN 978-0-521-59924-5.
  9. Sakarovitch, Jacques (2009), Elements of automata theory, Translated from the French by Reuben Thomas, Cambridge: Cambridge University Press, p. 26, ISBN 978-0-521-84425-3
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