In this thesis, we answer several questions relating to semigroup identities and tropical matrix semigroups. To begin, we look at two finiteness properties: weak permutability and strong permutability. We show that all tropical matrix semigroups are weakly permutable and that full and upper triangular tropical matrix semigroups are not strongly permutable for dimensions greater than 1. We then introduce and give a classification of truncated tropical semirings and fully describe which full matrix semigroups over truncated tropical semirings are strongly permutable. Next, we construct minimal and irredundant generating sets for upper triangular and unitriangular matrix semigroups over commutative semirings. We then give minimal and irredundant generating sets for the full matrix semigroup over the tropical integer semiring for dimensions 2 and 3, showing that the full matrix semigroup is finitely generated in dimension 2 but not in dimension 3. In addition to this, we construct finite presentations for upper triangular matrix semigroups over the tropical integers in every dimension. Turning towards the growth, we find new bounds on the degree of the polynomial growth of finitely generated subsemigroups of matrix semigroups over commutative bipotent semirings. In particular, for matrices over the tropical rational semiring, the bound of the degree of the polynomial growth is bounded only in the dimension of matrix semigroup, independent of the number of generators. We then explore the semigroup identities satisfied by tropical matrix semigroups and the plactic monoid of rank 4. We find a condition to show that a semigroup identity is not satisfied by the upper triangular tropical matrix semigroup of dimension n + 1, and use this to construct semigroup identities satisfied by the upper triangular tropical matrix semigroup of dimension n but not by dimension n+1. For full tropical matrix semigroups, we construct semigroup identities that are satisfied in dimension pâ1 but are not satisfied in dimension p for p prime. For the plactic monoid of rank 4, we find a new set of semigroup identities satisfied by the monoid, allowing us to deduce that the plactic monoid of rank 4 generates a different semigroup variety than the semigroup of upper triangular tropical matrices of dimension 5. In the final chapter, we construct a faithful representation of the stylic monoid of rank n by unitriangular tropical matrices of dimension n+1. We then show that the stylic monoid of rank n satisfies the exact same semigroup identities as the semigroup of unitriangular tropical matrices of dimension n + 1. Next, we consider involution semigroups, showing that the faithful morphism extends to involution semigroups. We show that the stylic monoid of rank n with involution is finitely based if and only n = 1. Finally, we show that, in contrast to the noninvolution case, the stylic monoid of rank n with involution and the semigroup of unitriangular tropical matrices of dimension n + 1 with involution satisfy different involution semigroup identities.
Date of Award  31 Dec 2023 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Marianne Johnson (Supervisor) & Mark Kambites (Supervisor) 

Semigroup identities of tropical matrix semigroups
Aird, T. (Author). 31 Dec 2023
Student thesis: Phd