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Re: abelian square
Posted:
Sep 8, 2013 2:50 PM


On Sunday, September 8, 2013 5:16:46 AM UTC7, Dieter von Holten wrote: > hi folks, > > i found 'abelian squares' in the context of patterns of n symbols of length 2n like 'abcd.cdba' where 'cdba' is a permutation of the first half 'abcd'. > > however, in those texts it is not explained (maybe just not clear enough for me) what is 'abelian' and where is the 'square' ??
http://www.wolframscience.com/conference/2004/presentations/HTMLLinks/index_35.html
<quote> The systematic study of structures in words was started by Axel Thue in 1906. In 1961, Paul Erdös raised the question whether abelian squares can be avoided in infinitely long words. An abelian square means a nonempty word uv, where u and v are permutations of each other. For example, abc acb is an abelian square. A word is called abelian squarefree if it does not contain any abelian square as a factor. For example, the word abacaba is abelian squarefree, while ab cabdc bcacd ac is not. </quote>
From "Abelian SquareFree Partial Words" by Francine BlanchetSadri, Jane I. Kim, Robert Merca?, William Severa, Sean Simmons <quote> Words or strings belong to the very basic objects in theoretical computer science. The systematic study of word structures (combinatorics on words) was started by a Norwegian mathematician Axel Thue at the beginning of the last century. One of the remarkable discoveries made by Thue is that the consecutive repetitions of nonempty factors (squares) can be avoided in infinite words overa threeletter alphabet. Recall that an infinite word w over an alphabet is said to be kfree if there exists no word x such that xk is a factor of w. For simplicity, a word that is 2free is said to be squarefree. Erdos raised the question whether abelian squares can be avoided in infinitely long words, i.e., whether there exist infinite abelian squarefree words over a given alphabet. An abelian square is a nonempty word uv, where u and v are permutations of each other. For example, abcacb is an abelian square. A word is called abelian squarefree, if it does not contain any abelian square as a factor. For example, the word abacaba is abelian squarefree, while abcdadcada is not (it contains the subword cdadca). </quote>
Note that in group theory, a word is any written product of group elements and their inverses http://en.wikipedia.org/wiki/Word_%28group_theory%29



