Related papers: On the semigroup of square matrices
This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question…
We show that the rank r component of the free idempotent generated semigroup of the biordered set of the full linear monoid of n x n matrices over a division ring Q has maximal subgroup isomorphic to the general linear group GL_r(Q), where…
We prove that, in both real and complex cases, there exists a pair of matrices that generates a dense subsemigroup of the set of $n\times n$ matrices.
We say that a semigroup of matrices has a submultiplicative spectrum if the spectrum of the product of any two elements of the semigroup is contained in the product of the two spectra in question (as sets). In this note we explore an…
We give examples of $n \times n$ matrices $A$ and $B$ over the filed $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ such that for almost every column vector $x \in \mathbb{K}^n$, the orbit of $x$ under the action of the semigroup generated by $A$…
We continue our study of exponent semigroups of rational matrices. Our main result is that the matricial dimension of a numerical semigroup is at most its multiplicity (the least generator), greatly improving upon the previous upper bound…
Every mathematician is familiar with the beautiful structure of finite commutative groups. What is less well known is that finite commutative semigroups also have a neat and well-described structure. We prove this in an efficient fashion.…
In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups $K_0(S)$ and $SK_0(S)$ of the isomorphism classes of the finitely generated projective and…
We study varieties of semigroups related to completely 0-simple semigroup. We present here an algorithmic descriptions of these varieties interms of "forbidden" semigroups.
Let $f\_1,\ldots, f\_s$ be formal power series (respectively polynomials) in thevariable $x$. We study the semigroup of orders of the formal series inthe algebra $K[[ f1,\ldots, f\_s]] \subseteq K[[ x ]]$ (respectively the semigroup of…
Let $n$ be a positive integer and $\mathcal M$ a set of rational $n \times n$-matrices such that $\mathcal M$ generates a finite multiplicative semigroup. We show that any matrix in the semigroup is a product of matrices in $\mathcal M$…
We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical algorithm to test nilpotency of…
Nilpotent semigroups in the sense of Mal'cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, $\mathsf{MN}$, which has finite rank. The semigroup identities that define nilpotent semigroups, lead…
Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to…
It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for…
We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X=N of natural numbers containing a given subsemigroup W of T(X) where each element of a given set $U$ is a generator of T(X) modulo W. This note…
We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation…
A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…
We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of…
Let $[n]=\{1,\ldots,n\}$ be the $n$-chain. We give presentations for the following transformation semigroups: the semigroup of full order-decreasing mappings of $[n]$, the semigroup of partial one-to-one order-decreasing mappings of $[n]$,…