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Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 62 / Issue 1 / March 2019
- Published online by Cambridge University Press:
- 07 January 2019, pp. 199-208
- Print publication:
- March 2019
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- Article
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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.
