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Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes

Part of: Theory of computing Semigroups

Published online by Cambridge University Press:  07 January 2019

Samuel J. van Gool  and
Benjamin Steinberg
Samuel J. van Gool
Affiliation:
Department of Mathematics, City College of New York, New York, New York 10031, USA Email: [email protected]@ccny.cuny.edu
Benjamin Steinberg
Affiliation:
Department of Mathematics, City College of New York, New York, New York 10031, USA Email: [email protected]@ccny.cuny.edu
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Abstract

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This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_{1}$, $T_{2}$, which together generate $T$, and the subsemigroup generated by their setwise product $T_{1}T_{2}$. In this sense we decompose $T$ by merging the subsemigroups $T_{1}$ and $T_{2}$. More generally, our technique merges semigroup homomorphisms from free semigroups.

Keywords

Krohn-Rhodes theoremaperiodic pointlikes

MSC classification

Primary: 20M07: Varieties and pseudovarieties of semigroups
Secondary: 20M35: Semigroups in automata theory, linguistics, etc. 68Q70: Algebraic theory of languages and automata
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 199 - 208
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Author S. J. v. G. was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant #655941. Author B. S. was supported by United States–Israel Binational Science Foundation #2012080 and NSA MSP #H98230-16-1-0047.

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