scholarly journals The ideal structure of idempotent-generated transformation semigroups

1985 ◽  
Vol 28 (3) ◽  
pp. 319-331 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Boolean Ring ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Ideal

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

Linear Transformations on Symmetric Spaces II

1993 ◽  
Vol 45 (2) ◽  
pp. 357-368 ◽  
Author(s):  
Ming–Huat Lim
Keyword(s):  
Vector Space ◽  
Product Space ◽  
Symmetric Spaces ◽  
Real Field ◽  
Dimensional Vector ◽  
Infinite Field ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

AbstractLet U be a finite dimensional vector space over an infinite field F. Let U(r) denote the r–th symmetric product space over U. Let T: U(r) → U(s) be a linear transformation which sends nonzero decomposable elements to nonzero decomposable elements. Let dim U ≥ s + 1. Then we obtain the structure of T for the following cases: (I) F is algebraically closed, (II) F is the real field, and (III) T is injective.

scholarly journals Invariants of Certain Groups I

1971 ◽  
Vol 41 ◽  
pp. 69-73 ◽  
Author(s):  
Takehiko Miyata
Keyword(s):  
Vector Space ◽  
Symmetric Algebra ◽  
Transcendence Degree ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Rational Field ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Natural Inclusion

Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K be the field of fractions of the symmetric algebra S(V) of V, then G acts naturally on K as k-automorphisms. There is a natural inclusion map V→K, so we view V as a k-subvector space of K. Let v1, v2, · · ·, vn be a basis for V, then K is generated by v1, v2, · · ·, vn over k as a field and these are algebraically independent over k, that is, K is a rational field over k with the transcendence degree n. All elements of K fixed by G form a subfield of K. We denote this subfield by KG.

Rank 1 preservers on the unitary Lie ring

1990 ◽  
Vol 49 (3) ◽  
pp. 399-417 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Lie Ring ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Commuting Pairs ◽  
Additive Maps

AbstractThe surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.

The Ideal Structure of Semigroups of Linear Transformations with Lower Bounds on Their Nullity or Defect

2010 ◽  
Vol 17 (01) ◽  
pp. 109-120 ◽  
Author(s):  
Suzana Mendes-Gonçalves ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Left Ideal ◽  
Lower Bounds ◽  
Dimensional Vector ◽  
Ideal Structure ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Maximal Regular Subsemigroup ◽  
The Ideal

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 dim V = 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 Green's 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).

scholarly journals Linear Map of Matrices

2008 ◽  
Vol 16 (3) ◽  
pp. 269-275 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Linear Map ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Matrix

Linear Map of MatricesThe paper is concerned with a generalization of concepts introduced in [13], i.e. introduced are matrices of linear transformations over a finitedimensional vector space. Introduced are linear transformations over a finitedimensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations over a finite-dimensional vector space is the same as the rank of the matrix of that transformation.

scholarly journals Sets of idempotents that generate the semigroup of singular endomorphisms of a finite-dimensional vector space

1982 ◽  
Vol 25 (2) ◽  
pp. 133-139 ◽  
Author(s):  
R. J. H. Dawlings
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Mathematical System

IfMis a mathematical system and EndMis the set of singular endomorphisms ofM, then EndMforms a semigroup under composition of mappings. A number of papers have been written to determine the subsemigroupSMof EndMgenerated by the idempotentsEMof EndMfor different systemsM. The first of these was by J. M. Howie [4]; here the case ofMbeing an unstructured setXwas considered. Howie showed that ifXis finite, then EndX=Sx.

On Some Invariants of a Bilinear Form

1961 ◽  
Vol 4 (3) ◽  
pp. 261-264
Author(s):  
Jonathan Wild
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Special Cases ◽  
Orthogonal Subspace

Let E be a finite dimensional vector space over an arbitrary field. In E a bilinear form is given. It associates with every sub s pa ce V its right orthogonal sub space V* and its left orthogonal subspace *V. In general we cannot expect that dim V* = dim *V. However this relation will hold in some interesting special cases.

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