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Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes
Published online by Cambridge University Press: 07 January 2019
Abstract
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.
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- © 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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