On Tropical Matrix Semigroups

Abstract

In this thesis, we study the congruences and subsemigroups of square tropical matrix semigroups. We begin by completely classifying the bicyclic subsemigroups and the congruences of the semigroup of two-by-two upper triangular tropical matrices. We do the latter by demonstrating that every congruence can be expressed using one of six explicit families of congruences. Then we completely classify the congruences of the semigroup of two-by-two full tropical matrices, demonstrating that every congruence falls into one of four explicit families of congruences. Finally we study the monogenic subsemigroups of the semigroup of n-by-n matrices over a semiring with properties that are realised by the tropical semiring. We prove various results about the period and index of these subsemigroups and give a complete classification of the monogenic subsemigroups of n-by-n tropical matrices for n less than or equal to 4.

Date of Award	14 Feb 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Marianne Johnson (Main Supervisor) & Mark Kambites (Co Supervisor)