Pseudo-solutions of word equations
Uploaded by: Sergey Kirgizov
Upload date: 2020-08-24 20:40:11
Edited at: 2020-08-24 22:37:55
Edited by: Sergey Kirgizov

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A famous example of the periodicity forcing word equation is $xy = yx$. It means that if a word $xy$ equals to a word $yx$, obtained by a _cut-and-swap_ process, there exists a word $w$ and integers $i,j$ such that $x = w^i$ and $y = w^j$. The paper's Section 1 and 5 contain a brief yet very helpful overview of existing results about periodicity forcing. About ten years ago, Czeizler, Kari and Seki, in their work [On a special class of primitive words](https://www.csd.uwo.ca/~lkari/mfcs.pdf), have introduced a notion of a pseudo-repetition. It can be explained with the following example. When DNA molecules are regarded as words, the Watson-Crick correspondence $$G \sim C, A \sim T$$ between guanine ($G$) and cytosine ($C$), adenine ($A$) and thymine ($T$) allows us to say that a word $ATCGATTGAGCTCTAGCG$ contains the same information as the word $TAGCTAACTCGAGATCGC$. A pseudo-repetition is a repetition of a word or its Watson-Crick complement. So, for instance $ATC \sim TAG$, and the word $ATCTAGATCATCTAG$ is a pseudo-repetition of the word $ATC$. In an abstract algebraic setting (involving monoids, morphic permutations, equivalence classes and anticongruences), the author studies an extension to the notion of periodicity forcing, using the recently introduced concept of pseudo-repetitions and related ideas. Holub proves, for instance, that any periodicity forcing classical word equation also forces a pseudo-periodicity. He provides interesting new results and generalizations of already known results. From MathSciNet MR4074724.
Sergey Kirgizov at 2020-08-24 22:37:19
Edited by Sergey Kirgizov at 2020-08-25 00:37:39

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