Finiteness properties for semigroups and their substructures

Lubbock, Diane (2021) Finiteness properties for semigroups and their substructures. Doctoral thesis, University of East Anglia.

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In this thesis we consider finiteness properties of infinite semigroups and infinite monoids. In particular we investigate finite presentations which have the property finite derivation type (FDT) or the property that they admit a presentation by a finite complete rewriting system (FCRS). We ask the question of whether these properties are inherited between a semigroup (or monoid) and particular substructures like subsemigroups (or submonoids).

We first investigate completely simple semigroups (which are isomorphic to Rees matrix semigroups) that have a single R-class or a single L-class. We prove that the maximal subgroups admit a presentation by a FCRS if and only if the semigroup admits a presentation by a FCRS with respect to a sparse generating set. Next we move on to our second stream of research and consider the property that a presentation has FDT. We study unitary subsemigroups with finite strict boundary (a condition given in terms of the Cayley graph) and prove that such subsemigroups inherit the property of FDT.

We prove that every finitely generated subsemigroup of the Bicyclic monoid admits a presentation by a FCRS. Finally we investigate FDT and FCRS for finitely generated submonoids of Plactic monoids, proving that these properties are satisfied in several cases. We make use of the fact that the Plactic monoid is known for having elements which correspond to semistandard tableau.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Chris White
Date Deposited: 21 Apr 2021 13:25
Last Modified: 21 Apr 2021 13:25

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