arXiv subjects

Thomas Aird

Publications and source records attributed to Thomas Aird.

10 recordsLinked to original sources

arXiv2025

Growth and identities of monogenic free adequate monoids

Motivated by recent advances in inverse semigroup theory, we investigate the growth of and identities satisfied by free left and free two-sided adequate monoids. We explicitly compute the growth of the monogenic free left adequate monoid with the usual unary monoid generating set and show it has intermediate growth owing to a connection with integer partitions. In the two-sided case, we establish a lower bound on the (idempotent) growth rate of the monogenic free adequate monoid, showing that it grows exponentially. We completely classify the enriched identities satisfied by the monogenic free left adequate monoid and deduce that it satisfies the same monoid identities as the sylvester monoid. In contrast, we show that the monogenic free two-sided adequate monoid satisfies no non-trivial monoid identities.

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arXiv2025

The Sch\"utzenberger groups and maximal subgroups of tropical matrices

We classify the Sch\"utzenberger groups of the category of matrices over the tropical semiring, $M(\mathbb{T})$, in doing so, we obtain a classification for the Sch\"utzenberger groups of the semigroupoid of matrices over the finitary tropical semiring, $M(\mathbb{F}\mathbb{T})$. We then classify the maximal subgroups of the monoid of $n \times n$ matrices over the tropical semiring, $M_n(\mathbb{T})$, for all $n \in \mathbb{N}$; generalising a result in the literature and correcting an erroneous proof. We proceed to show that for some $n \in \mathbb{N}$ there exists a group which appears as a Sch\"utzenberger group of $M_n(\mathbb{T})$ but does not appear as a maximal subgroup.

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arXiv2024

Short presentations for transformation monoids

By the theorems of Cayley and Vagner-Preston, the full transformation monoids and the symmetric inverse monoids play analogous roles in the theory of monoids and inverse monoids, as the symmetric groups do in the theory of groups. Every presentation for the finite full transformation monoids $T_n$, symmetric inverse monoids $I_n$, and partial transformation monoids $PT_n$ contains a monoid presentation for the symmetric group. In this paper we show that the number of relations required, in addition to those for the symmetric group, for each of these monoids is at least $4$, $3$, and $8$, respectively. We also give presentations for: $T_n$ with $4$ additional relations when $n\geq 7$; for $I_n$ with $3$ additional relations for all $n \geq 3$; and for $PT_n$ with $8$ additional relations for all $n\geq 7$. The presentations for $T_n$ and $I_n$ answer open problems in the literature.

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arXiv2023

Lattices of varieties of plactic-like monoids

We study the equational theories and bases of meets and joins of several varieties of plactic-like monoids. Using those results, we construct sublattices of the lattice of varieties of monoids, generated by said varieties. We calculate the axiomatic ranks of their elements, obtain plactic-like congruences whose corresponding factor monoids generate varieties in the lattice, and determine which varieties are joins of the variety of commutative monoids and a finitely generated variety. We also show that the hyposylvester and metasylvester monoids generate the same variety as the sylvester monoid.

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arXiv2023

Plactic-like monoids arising from meets and joins of stalactic and taiga congruences

We study the four plactic-like monoids that arise by taking the meets and joins of stalactic and taiga congruences. We obtain the combinatorial objects associated with the meet monoids, establishing Robinson-Schensted-like correspondences and giving extraction and iterative insertion algorithms for these objects. We then obtain results on the sizes of classes of words equal in plactic-like monoids, show that some of these monoids are syntactic, and characterise their equational theories.

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arXiv2023

Semigroup identities and varieties of plactic monoids

We study the semigroup identities satisfied by finite rank plactic monoids. We find a new set of semigroup identities of the plactic monoid of rank $n$ for $n \geq 4$, which are shorter than those previously known when $n \geq 6$. Using these semigroup identities we show that for all $n \in \mathbb{N}$, the plactic monoid of rank $n$ satisfies a semigroup identity not satisfied by the semigroup of $(n+1) \times (n+1)$ upper triangular tropical matrices. We then prove that the plactic monoid of rank $n$ generates a different semigroup variety for each rank $n$.

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arXiv2022

Tropical Representations and Identities of the Stylic Monoid

We exhibit a faithful representation of the stylic monoid of every finite rank as a monoid of upper unitriangular matrices over the tropical semiring. Thus, we show that the stylic monoid of finite rank $n$ generates the pseudovariety $\boldsymbol{\mathcal{J}}_n$, which corresponds to the class of all piecewise testable languages of height $n$, in the framework of Eilenberg's correspondence. From this, we obtain the equational theory of the stylic monoids of finite rank, show that they are finitely based if and only if $n \leq 3$, and that their identity checking problem is decidable in linearithmic time. We also establish connections between the stylic monoids and other plactic-like monoids, and solve the finite basis problem for the stylic monoid with involution.

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arXiv2022

Generating sets, presentations, and growth of tropical matrix monoids

We construct minimal and irredundant generating sets for a family of submonoids of the monoid of $n \times n$ upper triangular matrices over a commutative semiring. We show that the monoid of $n \times n$ matrices over the tropical integers, $M_n(\mathbb{Z}_\mathrm{max})$, is finitely generated if and only if $n \leq 2$, and finitely presented if and only if $n = 1$. Minimal and irredundant generating sets are explicitly constructed when $n \leq 3$. We then construct a presentation for the monoid of $n \times n$ upper triangular matrices over the tropical integers, $UT_n(\mathbb{Z}_\mathrm{max})$, demonstrating that it is finitely presented for all $n \in \mathbb{N}$. Finally, we establish upper bounds on the polynomial degree of the growth function of finitely generated subsemigroups of the monoid of $n \times n$ matrices over a bipotent semiring and show that these bounds are sharp for the tropical semiring.

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arXiv2021

Identities of tropical matrices and plactic monoids

We study semigroup varieties generated by full and upper triangular tropical matrix semigroups and the plactic monoid of rank 4. We prove that the upper triangular tropical matrix semigroup $UT_n(\mathbb{T})$ generates a different semigroup variety for each dimension $n$. We show a weaker version of this fact for the full matrix semigroup: full tropical matrix semigroups of different prime dimensions generate different semigroup varieties. For the plactic monoid of rank 4, $\mathbb{P}_4$, we find a new set of identities satisfied by $\mathbb{P}_4$ shorter than those previously known, and show that the semigroup variety generated by $\mathbb{P}_4$ is strictly contained in the variety generated by $UT_5(\mathbb{T})$.

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arXiv2021

Permutability of Matrices over Bipotent Semirings

We study permutability properties of matrix semigroups over commutative bipotent semirings (of which the best-known example is the tropical semiring). We prove that every such semigroup is weakly permutable (a result previous stated in the literature, but with an erroneous proof) and then proceed to study in depth the question of when they are strongly permutable (which turns out to depend heavily on the semiring). Along the way we classify monogenic bipotent semirings and describe all isomorphisms between truncated tropical semirings.

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