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On straight words and minimal permutators in finite transformation semigroups. (English) Zbl 1297.68173
Domaratzki, Michael (ed.) et al., Implementation and application of automata. 15th international conference, CIAA 2010, Winnipeg, MB, Canada, August 12–15, 2010. Revised selected papers. Berlin: Springer (ISBN 978-3-642-18097-2/pbk). Lecture Notes in Computer Science 6482, 115-124 (2011).
Summary: Motivated by issues arising in computer science, we investigate the loop-free paths from the identity transformation and corresponding straight words in the Cayley graph of a finite transformation semigroup with a fixed generator set. Of special interest are words that permute a given subset of the state set. Certain such words, called minimal permutators, are shown to comprise a code, and the straight ones comprise a finite code. Thus, words that permute a given subset are uniquely factorizable as products of the subset’s minimal permutators, and these can be further reduced to straight minimal permutators. This leads to insight into structure of local pools of reversibility in transformation semigroups in terms of the set of words permuting a given subset. These findings can be exploited in practical calculations for hierarchical decompositions of finite automata. As an example we consider groups arising in biological systems.
For the entire collection see [Zbl 1206.68008].

68Q70 Algebraic theory of languages and automata
20M20 Semigroups of transformations, relations, partitions, etc.
20M35 Semigroups in automata theory, linguistics, etc.
68R15 Combinatorics on words
GAP; Krohn-Rhodes; SgpDec
Full Text: DOI arXiv
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