Products of idempotent linear transformations

1985 ◽  
Vol 100 (1-2) ◽  
pp. 123-138 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

SynopsisIn 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

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.

scholarly journals Relative invariants and b-functions of prehomogeneous vector spaces

1985 ◽  
Vol 98 ◽  
pp. 139-156 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Linear Algebraic Group ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Rational Representation ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Prehomogeneous Vector Spaces ◽  
Relative Invariants

Let G be a connected linear algebraic group, p a rational representation of G on a finite-dimensional vector space V, all defined over C.

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 On nonzero component graph of vector spaces over finite fields

2017 ◽  
Vol 16 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Angsuman Das
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Clique Number ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Nonzero Component ◽  
Component Graph

In this paper, we study nonzero component graph [Formula: see text] of a finite-dimensional vector space [Formula: see text] over a finite field [Formula: see text]. We show that the graph is Hamiltonian and not Eulerian. We also characterize the maximal cliques in [Formula: see text] and show that there exists two classes of maximal cliques in [Formula: see text]. We also find the exact clique number of [Formula: see text] for some particular cases. Moreover, we provide some results on size, edge-connectivity and chromatic number of [Formula: see text].

Products of idempotent endomorphisms of an independence algebra of infinite rank

1993 ◽  
Vol 114 (2) ◽  
pp. 303-319 ◽  
Author(s):  
John Fountain ◽  
Andrew Lewin
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Common Generalization ◽  
Finite Dimensional Vector Space ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Linear Maps

AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

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.

scholarly journals The functional equation of zeta distributions associated with prehomogeneous vector spaces

1985 ◽  
Vol 99 ◽  
pp. 131-146 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Functional Equation ◽  
Linear Algebraic Group ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Rational Representation ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Prehomogeneous Vector Spaces ◽  
Complex Number Field

Let (G, ρ, V) be a triple of a linear algebraic group G and a rational representation ρ on a finite dimensional vector space V, all defined over the complex number field C.

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.

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