Equations on Partial Words

F. Blanchet-Sadri, D. Dakota Blair, and Rebeca V. Lewis


It is well known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation x m y n = z p has only periodic solutions in a free monoid, that is, if x m y n = z p holds with integers m, n, p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot of attention, was first proved by Lyndon and Schützenberger for free groups.

In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of “do not know” symbols. When we speak about equations on partial words, we replace the notion of equality (=) with compatibility (↑). Among other equations, we solve xyyx, xzzy, and x m y nz p for integers m, n, p ≥ 2.

Keywords: Equations on words, equations on partial words, commutativity, conjugacy, free monoid.

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Acknowledgement:   This material is based upon work supported by the National Science Foundation under Grant No. DMS-0452020 .

Disclaimer:   Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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