Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes. Issue 1 (7th January 2019)

• Record Type: Journal Article
• Authors: van Gool, Samuel J.; Steinberg, Benjamin
• Is Part Of: Canadian mathematical bulletin =. Volume 62:Issue 1(2019)
• Page Start: 199
• Page End: 208
• Publication Date: 2019-01-07
• Subjects: 20M07, -- 20M35, -- 68Q70; Krohn-Rhodes theorem, -- aperiodic pointlikes; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2018-014-8
• Languages: English
• ISSNs: 0008-4395

Abstract

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.