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

Gool, Samuel J. van ; Steinberg, Benjamin

Canadian mathematical bulletin, Tome 62 (2019) no. 1, pp. 199-208

Résumé

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.

Détail

DOI : 10.4153/CMB-2018-014-8

Mots-clés : Krohn-Rhodes theorem, aperiodic pointlikes