Semigroup Forum

, Volume 68, Issue 3, pp 365–372 | Cite as

Zebra Factorizations in Free Semigroups

Research Article
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Abstract

Let $S$ be a semigroup of words over an alphabet $A$. Let $\Omega(S)$ consist of those elements $w$ of $S$ for which every prefix and suffix of $w$ belongs to $S$. We show that $\Omega(S)$ is a free semigroup. Moreover, $S$ is called separative if also the complement $S^c = A^+\setminus S$ is a semigroup. There are uncountably many separative semigroups over $A$, if $A$ has at least two letters. We prove that if $S$ is separative, then every word $w \in A^+$ has a unique minimum factorization $w = z_1z_2 \cdots z_n$ with respect to $\Omega(S)$ and $\Omega(S^c)$, where $z_i \in \Omega(S) \cup \Omega(S^c)$ and $n$ is as small as possible.

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Copyright information

© Springer-Verlag New York Inc. 2004

Authors and Affiliations

  1. 1.Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309-0347USA
  2. 2.Department of Mathematics, University of Turku, FIN-20014 TurkuFinland
  3. 3.Leiden Institute for Advanced Computer Science, Leiden University, Niels Bohrweg 1 2333 CA Leiden, The Netherlands and Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309-0347USA

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