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

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 decompo...

Full description

Saved in:
Bibliographic Details
Published in:Canadian mathematical bulletin 2019-03, Vol.62 (1), p.199-208
Main Authors: van Gool, Samuel J., Steinberg, Benjamin
Format: Article
Language:English
Subjects:
Decomposition
Homomorphisms
Theorems
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0008-4395
1496-4287
DOI:10.4153/CMB-2018-014-8