THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

scholarly journals A Norm Inequality for Linear Transformations

1961 ◽  
Vol 4 (3) ◽  
pp. 239-242
Author(s):  
B.N. Moyls ◽  
N.A. Khan
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Hermitian Operator ◽  
Norm Inequality ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Image Position ◽  
Ky Fan

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then1

Solvable and Nilpotent Subgroups of GL(n,qm)

1982 ◽  
Vol 34 (5) ◽  
pp. 1097-1111 ◽  
Author(s):  
Thomas R. Wolf
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Solvable Subgroup ◽  
Linear Transformations ◽  
Completely Reducible ◽  
Image Position

Let V ≠ 0 be a vector space of dimension n over a finite field of order qm for a prime q. Of course, GL(n, qm) denotes the group of -linear transformations of V. With few exceptions, GL(n, qm) is non-solvable. How large can a solvable subgroup of GL(n, qm) be? The order of a Sylow-q-subgroup Q of GL(n, qm) is easily computed. But Q cannot act irreducibly nor completely reducibly on V.Suppose that G is a solvable, completely reducible subgroup of GL(n, qm). Huppert ([9], Satz 13, Satz 14) bounds the order of a Sylow-q-subgroup of G, and Dixon ([5], Corollary 1) improves Huppert's bound. Here, we show that |G| ≦ q3nm = |V|3. In fact, we show thatwhere

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 Real Subspaces of a Quaternion Vector Space

1978 ◽  
Vol 30 (6) ◽  
pp. 1228-1242 ◽  
Author(s):  
Vlastimil Dlab ◽  
Claus Michael Ringel
Keyword(s):  
Vector Space ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Image Position ◽  
Real Subspace

If UR is a real subspace of a finite dimensional vector space VC over the field C of complex numbers, then there exists a basis ﹛e1, … , en﹜ of VG such that

Anticommuting Linear Transformations

1961 ◽  
Vol 13 ◽  
pp. 614-624 ◽  
Author(s):  
H. Kestelman
Keyword(s):  
Quantum Mechanics ◽  
Vector Space ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Pauli Matrices ◽  
Image Position

It is well known that any set of four anticommuting involutions (see §2) in a four-dimensional vector space can be represented by the Dirac matrices(1)where the B1,r are the Pauli matrices(2)(See (1) for a general exposition with applications to Quantum Mechanics.) One formulation, which we shall call the Dirac-Pauli theorem (2; 3; 1), isTheorem 1. If M1,M2, M3, M4 are 4 X 4 matrices satisfyingthen there is a matrix T such thatand T is unique apart from an arbitrary numerical multiplier.

scholarly journals On the Index of a Symmetric Form

1960 ◽  
Vol 3 (3) ◽  
pp. 293-295
Author(s):  
Jonathan Wild
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Dimensional Vector ◽  
Symmetric Form ◽  
Characteristic P ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let E be a finite dimensional vector space over a finite field of characteristic p > 0; dim E = n. Let (x,y) be a symmetric bilinear form in E. The radical Eo of this form is the subspace consisting of all the vectors x which satisfy (x,y) = 0 for every y ϵ E. The rank r of our form is the codimension of the radical.

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 Combinatorial polarization, code loops, and codes of high level

2004 ◽  
Vol 2004 (29) ◽  
pp. 1533-1541 ◽  
Author(s):  
Petr Vojtechovský
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Binary Codes ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Moufang Loops ◽  
Hamming Weights ◽  
High Level

We first find the combinatorial degree of any mapf:V→F, whereFis a finite field andVis a finite-dimensional vector space overF. We then simplify and generalize a certain construction, due to Chein and Goodaire, that was used in characterizing code loops as finite Moufang loops that possess at most two squares. The construction yields binary codes of high divisibility level with prescribed Hamming weights of intersections of codewords.

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