Suzana Mendes-Gonçalves

The ideal structure of semigroups of transformations with restricted range

Let Y be a fixed nonempty subset of a set X and let T (X, Y ) denote the semigroup of all total transformations from X into Y . In 1975, Symons described the automorphisms of T (X, Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T (X, Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T (X, Y ) when |Y | 6= 1 and Y 6= X ; and using this, they described the Green’s relations on T (X, Y ). Here, we use their work to describe the ideal structure of T (X, Y ). We also correct the proof of the corresponding result for a linear analogue of T (X, Y ). 2010 Mathematics subject classification: primary 20M20; secondary 15A04.

Semigroups of transformations restricted by an equivalence

Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α−1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.

Baer–Levi semigroups of linear transformations

Given an infinite-dimensional vector space V , we consider the semigroup GS(m,n) consisting of all injective linear α : V → V for which codim ranα = n where dimV = m ≥ n ≥ א0. This is a linear version of the well-known Baer-Levi semigroup BL(p, q) defined on an infinite set X where |X| = p ≥ q ≥ א0. We show that, although the basic properties of GS(m,n) are the same as those of BL(p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS(m,n) and some of its maximal subsemigroups: in this, we follow previous work on BL(p, q) by Sutov (1966) and Sullivan (1978) as well as Levi and Wood (1984).

Maximal Inverse Subsemigroups of the Symmetric Inverse Semigroup on a Finite-Dimensional Vector Space

ABSTRACT Yang (1999) classified the maximal inverse subsemigroups of all the ideals of the symmetric inverse semigroup I(X) defined on a finite set X. Here we do the same for the semigroup I(V) of all one-to-one partial linear transformations of a finite-dimensional vector space. We also show that I(X) is almost never isomorphic to I(V) for any set X and any vector space V, and prove that any inverse semigroup can be embedded in some I(V).

The ideal structure of semigroups of linear transformations with lower bounds on their nullity or defect

Suppose V is an infinite-dimensional vector space and let T(V) denote the semigroup (under composition) of all linear transformations of V. In this paper, we study the semigroup OM(p,q) consisting of all α ∈ T(V) for which dim ker α ≥ q and the semigroup OE(p,q) of all α ∈ T(V) for which codim ran α ≥ q, where dimV = p ≥ q ≥ ℵ0. It is not difficult to see that OM(p,q) and OE(p,q) are a right ideal and a left ideal of T(V), respectively, and using these facts, we show that they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Also, we describe Greens relations and the two-sided ideals of each semigroup, and determine its maximal regular subsemigroup. Finally, we determine some maximal right cancellative subsemigroups of OE(p,q).

Semigroups of Injective Linear Transformations with Infinite Defect

ABSTRACT Given an infinite-dimensional vector space V, we consider the semigroup KN(p, q) consisting of all injective linear transformations α : V → V, for which the codimension of the range of α is at least q, where dim V = p ≥ q ≥ ℵ0. Kemprasit and Namnak (2001) considered the semigroup KN(p, ℵ0) while deciding when certain subsemigroups of T(V )—the semigroup under composition of all linear transformations from V to V—belong to BQ—the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. In this article, we determine when KN(p, q) belongs to BQ in terms of the dimension of V. Next we characterize Greens relations on KN(p, q), and determine its one and two sided ideals; and we use this information to show that KN(p, q) is a model for certain types of algebraic semigroups. Then we describe all quasi-ideals and bi-ideals of KN(p, q). We also determine its maximal right simple subsemigroups. Communicated by D. Easdown.

Almost injective transformations of an infinite set

Let V be an infinite-dimensional vector space, let n be a cardinal such that ℵ0 ≤ n ≤ dim V, and let AM(V, n) denote the semigroup consisting of all linear transformations of V whose nullity is less than n. In recent work, Mendes-Gonçalves and Sullivan studied the ideal structure of AM(V, n). Here, we do the same for a similarly-defined semigroup AM(X, q) of transformations defined on an infinite set X. Although our results are clearly comparable with those already obtained for AM(V, n), we show that the two semigroups are never isomorphic.

ISOMORPHISM PROBLEMS FOR TRANSFORMATION SEMIGROUPS

Let X be an arbitrary set and let P (X) denote the set of all partial transformations of X : that is, all transformations α whose domain, dom α, and range, ranα, are subsets of X . As usual, the composition α ◦ β of α, β ∈ P (X) is the transformation with domain Y = (ranα ∩ dom β)α such that, for all x ∈ Y , x(α ◦ β) = (xα)β, and we write α ◦ β more simply as αβ. It is well-known that (P (X), ◦) is a semigroup. Let T (X) denote the subsemigroup of P (X) consisting of all α ∈ P (X) with domain X , and let I(X) denote the symmetric inverse semigroup on X : that is, the set of all injective elements of P (X). If α ∈ T (X), we define the rank of α to be r(α) = | ranα| and we define another two cardinal numbers, called the defect and the collapse of α, respectively, as follows.