R. P. Sullivan

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).

Factorisable Semigroups of Generalized Transformations

ABSTRACT If X and Y are sets, we let P(X, Y ) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). We define an operation * on P(X, Y ) by choosing θ ∈ P(Y, X) and writing: α*β = α °θ°β, for each α, β ∈ P(X, Y ). Then (P(X, Y ), *) is a semigroup, and some authors have determined when this is regular (Magill and Subbiah, 1975), when it contains a “proper dense subsemigroup” (Wasanawichit and Kemprasit, 2002) and when it is factorisable (Saengsura, 2001). In this paper, we extend the latter work to certain subsemigroups of (P(X, Y ), *). We also consider the corresponding idea for partial linear transformations from one vector space into another. In this way, we generalise known results for total transformations and for injective partial transformations between sets, and we establish new results for linear transformations between vector spaces.

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).

Products of nilpotent matrices

In 1975, Paul Halmos asked: when can a square matrix be written as a product of nilpotent matrices? This was answered indirectly by Sourour in 1992: if n ≥ 3 then any n × n singular matrix can be written in this way. In this article, we determine the best possible index for the nilpotents in such a product, and compare our work on linear transformations of a vector space with analogous results for transformations of a set.

Factorisable Semigroups of Linear Transformations

Let P(X) be the semigroup of all partial transformations of a set X. A subsemigroup S of P(X) is factorisable if S = GE = EH, where G, H are subgroups of S and E is the set of idempotents in S. In 2001, Jampachon, Saichalee and Sullivan proved a simple result that generalized most of the previous work on factorisable subsemigroups of P(X). They also determined when the semigroup T(V) of all linear transformations of a vector space V is factorisable. In this paper, we extend that work to partial linear transformations of V and consider the notion of locally factorisable for such semigroups.

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.