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Topic: abelian square
Replies: 5   Last Post: Sep 8, 2013 2:50 PM

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 FredJeffries@gmail.com Posts: 1,845 Registered: 11/29/07
Re: abelian square
Posted: Sep 8, 2013 2:50 PM

On Sunday, September 8, 2013 5:16:46 AM UTC-7, 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' ??

<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 non-empty word uv, where u and v are permutations of each
other. For example, abc acb is an abelian square. A word is called
abelian square-free if it does not contain any abelian square as
a factor. For example, the word abacaba is abelian square-free,
while ab cabdc bcacd ac is not.
</quote>

From "Abelian Square-Free Partial Words" by Francine Blanchet-Sadri,
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 non-empty factors
(squares) can be avoided in infinite words overa three-letter
alphabet. Recall that an infinite word w over an alphabet is said
to be k-free if there exists no word x such that xk is a factor of w.
For simplicity, a word that is 2-free is said to be square-free.
Erdos raised the question whether abelian squares can be avoided
in infinitely long words, i.e., whether there exist infinite
abelian square-free words over a given alphabet. An abelian square
is a non-empty word uv, where u and v are permutations of each other.
For example, abcacb is an abelian square. A word is called abelian
square-free, if it does not contain any abelian square as a
factor. For example, the word abacaba is abelian square-free, while
</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

Date Subject Author
9/8/13 Dieter von Holten
9/8/13 Jussi Piitulainen
9/8/13 Peter Percival
9/8/13 Peter Percival
9/8/13 fom
9/8/13 FredJeffries@gmail.com