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Representation of a Full Transformation Semi-group Over a Finite Field

Munir Ahmed1*, Muhammad Naseer Khan2, Muhammad Afzal Rana2

1Department of Mathematics, Islamabad Model College For Boys, Islamabad, Pakistan

2Department Of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan

Corresponding Author:
Munir Ahmed
Department Of Mathematics, Islamabad Model College For Boys, F-10/4, Islamabad, Pakistan
E-mail: irmunir@yahoo.com

Received date: 09/01/2018 Accepted date: 10/05/2018 Published date: 11/06/2018

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Abstract

In this paper we discuss the representations of a full transformation semigroup over a finite field. Furthermore, we observe some properties of irreducibility representation of a full transformation semigroup and discuss the linear representation of a zero-adjoined full transformation semigroup. Moreover, we characterize the linear representation of a full transformation semigroup over a finite field Fq (where q is a prime power) in terms of Maschke’s Theorem. Finally, we observe that there exists an isomorphism between the full matrix algebra (Fq)m and the space of all linear transformation L(Fq m) on an m-dimensional vector space Fq m

Keywords

Semigroup of transformations; Representation of semigroup; Finite field

Introduction

Serre has given a comprehensive theory of linear representation of finite groups in [1]. It has been obtained in the group theory that the number of simple FG− modules is equal to the number of conjugacy classes of the group G such that the characteristic of the field F does not divide the order of G. A lot of work is done for the classification of groups in terms of its representation and characterization.

By Clifford, each element of a semigroup is uniquely determined by a matrix over a field and a complete classification of the representations of a particular class of a semigroups is given in [2-4]. Moreover, irreducible representations of a semigroup over a field is obtained as the basic extensions to the semigroup of the extendible irreducible representations of a group, and the representations of completely simple semigroup is also constructed in [2-4].

Stoll has given a characterization of a transitive representation, and obtained a transitive representation of a finite simple semigroup, see [5]. The construction of all representations of a type of finite semigroup which is sum of a set of isomorphic groups is also obtained. Munn obtained a complete set of inequivalent representations of a semigroup S which are irreducible in terms of those of its basic groups of its principal factors. He also introduced the principal representations of a semigroup in [6]. A representation of semigroup whose algebra is semisimple is characterized in [7,8]. The representation of a finite semigroup for which the corresponding semigroup algebra is semisimple is also obtained. An explicit determination of all the irreducible representations of Tn is due to Hewit and Zuckerman in [9].

There is a one-to-one correspondence between the representations of a group G and the nonsingular representations of the semigroup S, which preserves equivalence, reduction and decomposition [10].

In the case of an irreducible representation of a finite semigroup, the factorization can be avoided and an explicit expression of such representation is given in [11]. We consider a full transformation semigroup Tn to obtain its combinatorial property with regard to its irreducible representations. There exists a non-zero linear transformation satisfying some specific conditions in Theorem 7.3.

It is observed that for the basis equation of a vector spaceequation , there is a natural one-to-one correspondence (between the resentations of a full transformation semigroupequation over a finite field Fq and those of the algebra Fq[equation] which preserves, equivalence, reduction and decomposition into irreducible constituents.

Consequently, we reinterpret the Maskhe Theorem [12] regarding the algebra Fq[equation] i.e., the algebra Fq[equation] is semisimple if and only if the characteristic of Fq does not divide the order mm of the full transformation semigroup equation.

The representation of full trasformation semigroup over a finite field is discussed in Section-8, specially the Maschke’s theorem is restated for the semisimplicity of the semigroup algebra Fq[equation], see Theorem 8.1 Finally, a linear algebraic result regarding the isomorphism between the full matrix algebra (Fq)m and the space of all the linear transformations on Fqm is given in Theorem 8.2.

Preliminaries

Definition

A transformation semigroup is a collection of maps of a set into itself which is closed under the operation of composition of functions. If it includes identity mapping, then it is a monoid. It is called a transformation monoid.

If (X,S) is a transformation semigroup then X can be made into semigroup action of S by evaluation, x.s=xs=y for s equation S, and x,y equationX. This is the monoid action of S on X, if S is a transformation monoid.

Hewitt and Zuckerman gives a treatment of the irreducible representation of the transformation semigroup on a set of finite cardinality [8]. The result for the case of a finite semigroup S with F[S] semisimple was given by Munn in [13].

The full reducibility and the proper extensions of irreducible representations of a group to those of a semigroup are the basic extensions.

THEOREM 2.2

Full reducibility holds for the representations of a semigroup S over the field F if and only if

Full reducibility holds for the extendible representations of G over F, and

The only proper extension of a proper representation of G to S is the basic extension [14].

A representation M of S is homomorphism of S into the multiplicative semigroup of all (α,α) matrices( where α is an arbitrary positive integer) such that M(x) ≠ 0 for some x equation S. If the set {M(x): x equation S} is irreducible i.e., if every (α,α) matrix is a linear combination of matrices M(x), then M is said to be an irreducible representation of S. The identity representation is the mapping that carries every x equation S into the identity matrix.

Full transformation semigroup

The idea of studying Tn was suggested by Miller (in oral communication). The problem of obtaining representations of semigroup as distinct from groups have been first studied by Suskevic. Clifford has given a construction of all representations of a class of semigroups closely connected with Tn. Ponizovski has pointed out some simple properties of Tn . In the present discussion, we relate the irreducible representations of Tn to that of its semigroup algebra L(Tn). The set of all transformations of set X into itself is called the full transformation semigroup under the binary operation of multiplication as the composition of transformation analogue of the symmetric group GX. Let Xn = {1,2,3,….,n} be a finite set and denote the semigroup TXn of all the self-maps of Xn into Xn. If cardinality of Xn is n, denote Tn for TXn then the cardinality of Tn is nn [15].

Example

The set S={e,a,x,y} is a semigroup under the multiplication. The Cayley’s multiplication table of S is given as follows [16].

equation

If the mapping equation is given byequation then ф embeds S in equation . It can also be seen that the map equation is defined by

equation

equation

and

equation

embeds S into equation

Notice that y is a right regular representation of S, whereequation as defined above (where ψ(e), ψ(a), ψ(x), ψ(y) equation TS) is such that for any sequationS, we have

equation

So ψ is a right regular representation of S.

Regular representation of a transformation semigroup

Let K denote the set of right zero elements of a semigroup S. Then, equation if and only if

(i) for all x in K, and all a,b in S, xa=xb implies a=b;

(ii) if α is any transformation of K, then there exists a in S such that xα = xa for all x equation K.

An element α of TX is idempotent if and only if it is the identity mapping when restricted to Xα. Suppose that X is a set of cardinality n. Then, the full transformation semigroup TX contains the symmetric group GX of degree n. equation then the rank r of α is defined by equation. and the defect of the element a is given by n-r. If b is an element of TX of rank r<n, then there exists elements γ and δ of TX such that g has the rank r+1, δ has the rank n-1, and β =γδ (we can choose δ as an idempotent, and γ different from β at only one part of X). By induction, every element of TX of defect k(1 ≤ k ≤ n-1) can be expressed as the product of an element of GX and k number of(idempotent) elements of defect 1, see also [17].

If α equation TX is of defect 1, then every other element of TS of defect 1 can be expressed in the form λαμ with λ and μ are in GX. If α is an element of TS of defect 1, then <GXα>= TS .

Let X=S be a semigroup, an element ρ equation TS is said to be a right translation of S if x(yρ) = (xy)ρ for all x,y equation S and λ equation TX is said to be a left translation of S if (xλ)y = (xy)λ for any x,y equation S. The left and a right translations λ and ρ, respectively, are called linked if x(yl) = (xr)y for all x;y 2 S.

Note that λaλ = λ and ρaρ = ρ, if λ and ρ are linked, then

equation

Let S = {e,f,g,α} be a semigroup with the operation “.” given by the Cayley’s table

equation

Cayley’s table

The transformation

equation

is a left translation which is not linked with any right translations of S. We recall the following proposition regarding the semisimple algebra.

Proposition

An algebra A is a semisimple if and only if A-module of A is semisimple.

Definition

Let S be a semisimple with zero element z. The contracted algebra F0[S] of S over F is an algebra over F containing a basis equation such that equation U0 is a subsemigroup of F0[S] isomorphic with S. A semisimple algebra can also be regarded as a contracted semigroup algebra.

We recall the following facts regarding the representations of a semisimple algebra.

Lemma

(a) Let equation be an algebra having finite order over the field F, and let equation be a radical ofequation . Then, every non-null irreducible representation of equation maps equation into 0, and so it is effectively a representation of the semisimple algebra equation/ equation.

(b) Let ф be any faithful representation of a semisimple algebra equation and let P be an n*n matrix over τ Then, P is nonsingular if and only if ф(n)(P) is non-singular [18].

THEOREM 4.4

(6, Th. 5.7). An irreducible algebra of linear transformations is simple.

If A equation (F)n, then the transformation x → Ax of a vector space V is linear transformation τ of V to V, and the mapping A → A is an isomorphism of (F)n upon the algebra L[TV] of all linear transformations of V. A homomorphism ф of equation into (F)n is called a representation of equation of degree n over F. In other words, to each element x of equation there corresponds an n*n matrix ф(x) such that

ф(x+y) = ф(x)+ф(y);

ф(xy) = ф(x)ф(y);

ф(αx) = αф(x):

for all x,y in TN and α in F.

The irreducible representations of semigroups

Let f be an element of TN . Then, f splits the set {1,2,..,n} into a number p of nonvoid disjoint subsets, each of the form {x:f(x)=a} for some a equation rang( f). Obviously, f is determined by these sets and the corresponding a's. For nonvoid subset s of {1,2,…,n}, let s* be the least element of s. Write the sets {x: f(x)=a} in the order s1,s2,..,sp where s*1<s*2<…< s* p, and represent f by the symbol

equation

where 1 equation n, the class of sets s1,…,sp is a decomposition of {1,2,..,n} of the kind described above, and a1,a2,…,ap are any distinct integers lying between 1 and n. The expression s1,..,sp will always mean a decomposition of {1,2,..,n} into nonvoid, disjoint subsets with s*1<s*2<…< s* p. The letters t and w will be used similarly. Also a1,a2,...,ap will always mean any ordered sequence of distinct integers from 1 to n; the letters c and d will be used similarly.

For p = 1,2,…,n, let equationp be the set of all elements of equationN whose range contains just p elements, that is,

equation

for a fixed p. Strictly speaking, equationN depends upon n as well as p. However, only one value of n will be treated at one time. The set equationN is obviously the symmetric group Sn. The set equationp 1 is a semigroup with the trivial multiplication fg=f. No other equationp is a subsemigroup of equationp. It will be convenient to have the semigroup equationp U{z}, with multiplication defined by

equation

Using a linear algebraic result, we have the following formula regarding the rank of a linear representation of Tn .

THEOREM 5.1

Let M be an irreducible linear representation of Tn , and let S={f: equation f Tn and M( f)=0}, then rank[M( Tn )]

equation

Proof

Suppose the irreducible linear representation M: Tn → L(Tn) is as given above. Since M is irreducible representation of Tn . Thus, using a result in, the set S is void or equation

Since,

equation

where F is a field of characteristic 0.

Since,

equation

and,

equation

We have

equation

Therefore,

This completes the proof.

Let X={x1,x2,…,xn} be a set of cardinality n and let Sn denote the set of all single-valued maps of X to itself. We have the following characterization of a map from Sn into the set of all n*n matrices Dn over the field F, see also.

THEOREM 5.2

Let M:Sn →Dn be a map defined by M(f) = Af equation Dn, for f equation Sn. Then, M forms a homomorphism of Sn into Dn. If, in particular, Sn is a semigroup S, then M becomes a representation of S ∪{z} into Dn (where z is a zero element).

Proof

For any two single valued maps f and g in Sn, the product fg is also a single valued map, therefore fg equation Sn.

Moreover, since equation In particular, if i is the identity map on X, thenequation then we have;

equation

Therefore, M defines a homomorphism of Sn into Dn.

If, in particular, if equation the semigroup of all maps from X into itself, then we can define an induced structure on the adjoined zero semigroup Tn , where z is a zero element, i.e., for any f Tn , we have equation

The induced structure on equation is defined as follows:

equation

Then, the homomorphism M can be extended into a map equation of the semigroupequation is defined by

equation

Therefore,

equation

And

equation

Thus, equation becomes a representation on S.

Representation of a semigroup of linear transformations in green’s

Relations

Two things that can be associated with an element α equationequation are as follows:

1. the range Xα of α, and

2. the partition equation if xα=yα which defines an equivalence relation on X.

Let equation be the natural mapping of X upon the set equation of equivalence classes of X modequation Then,equation becomes a one-to-one mapping of equation upon Xα. It follows that equation , and this cardinal number is called the rank of α.

Remark

The Ex.2.2.6 in [4] can be rewritten as follows,

Let F be a field and V be a vector space over F. By the dimension dimV of we mean the cardinal number of a basis of V over F. Let equation (V ) be the multiplicative semigroup (i.e., under the operation of composition of maps) of all linear transformations of V with each element t of L(V) we associate two subspaces of V that are given as follows:

1. the range equation , consisting of all (x) τ with x equation V and,

2. the null space Nτ of τ , consisting of all y in V such that (y)τ = 0.

(a) Let equation , and W be a subspace of V, complementary to the null space Nτ , so that V = τ

Then, τ induces a non-singular matrix A.

Hence, dim(V=Nτ )=dim(W)=dim(Vt); is called rank of t. The difference or quotient space of V modulo Nτ is denoted by V-Nτ or by V/N equation(Tv) . If dimV is finite, this notation of rank is the usual one as for the matrix A, since VA is the row-space of A. Also NA is the orthogonal complement of the column-space of A.

(b) Two elements of the space equation equivalent if and only if they have the same range (null-space). (c) If N and W are subspaces of V such that dim(V/Nτ )=dimW, then there exists at least one element ρ of 1τ such that N = Nρ and W =Vρ.

(d) Two elements equation equivalent if and only if rankequation

(e) The Th. 2.9 holds for equation(v) instead of TX if we replace “subset Y of X” by “the subspace W of V”, Tv by dim W, “partition Tv of X” by “subspace N of V ”, and X /Π by dim(V/N).

Linear representation of a full transformation semigroup over a finite field

Definition

Let V be a vector space over the field F(=C) the complex numbers and let the finite subset equation of V be a basis for V, i.e., dimV=n, let Tv denote the full transformation semigroup over V. The space equation(Tv) denotes the space of all linear transformations on V. If a is in equation(Tv) a linear transformations, then, each a:V→V is represented by a square matrix (aij) of order n. The coefficients aij are complex numbers for all i and j=1,…,n and are obtained by

equation

where a can be identified as a morphism which is equivalent to saying that det(a)=det(aij) ≠ 0. The linear space equation(Ts) of full transformation semigroup can be identified with the semigroup of all transformations of degree n.

A representation ф : S → equation(Ts) is faithfull if and only if ф is one-to-one homomorphism. A representation ф of a semigroup S, of degree n over the field F, we mean a homomorphism of S into the semigroup equation(TFn) of all linear transformation over Fn, where equation the vector space is generated by S over the field F. Thus, to each element s of S there corresponds a linear transformationequation such that

equation

We denote the algebra of all linear transformations over the n-dimensional vector space Fn over the field F by equation . Obviously,equation appears as a subspace ofequation

If ф is an isomorphism of S upon a subsemigroup of equation; then ф is said to be faithfull. We shall determine all the representations of various classes of finite semigroups over a finite field Fq. If S is a finite semigroup, then there is a one-to-one correspondence between a representation of S and that of algebra equation over the finite field Fq. Of course, this correspondence preserves the reducation, decomposition and hence the full reducibility hold for such representations of S if and only if equation is semisimple that holds if q does not divide the dimFnq=n, (the dimension of the vector space Fnq over a finite field Fq. There is a necessary and sufficient condition on a finite semigroup S in order that Fq[S] is semisimple. An explicit representation of such group is obtained in. They constructed all the irreducible representations of S from those of its principal factors of the full transformation semigroup on a finite set.

If F is algebraically closed, then there are no division algebras over F other than F itself, and in this case Wedderbun’s second theorem tells us that every simple algebra ∧ over F is isomorphic with the full transformation semigroup algebra ∧ of degree n for some positive integer n.

Any isomorphism of ∧ upon semigroup ∧ is a representation of ∧ , and gives the irreducible representation of ∧ . Let ∧ be an algebra of order n over F, and let ф be a representation of equation of degree r over F, and let m be a positive integer. For each element ф(m) of equation, construct a transformationequation

such that

equation

equation

if

equation

then

The map ф(m) is called the representation of L(Lm) associated with the representation ф of ∧ . The following lemma is due to Van der Waerden’s modern algebra.

Lemma

Let D be division algebra, and let m be a positive integer. The right regular representation ρ of D is an irreducible, and the only irreducible representation of the simple algebra equation(Dm ) is just the representation ρ(m) of equation(Dm) associated with ρ.

THEOREM 7.3

Let ∧ σ (σ=1,…,c) be the simple components of a semisimple algebra ∧ . By Wedderburn’s second theorem, each equation σ may be regarded as a full transformation equation of some degree mσ over the division algebra equation(∧σ ) . Let ρσ be the regular representation of Dσ and ρσ(mσ) be the representation of equation (∧σ ) associated with ρσ then ρσ(mσ) is the only irreducible representation of ρσ. Extending (ρσ)(mσ) be the representation of equation (∧σ ) s sociated with ρσ then ρσ(mσ) is the only irreducible representation of ρσ. Extending (ρσ)(mσ) to equation by defining фσ(a) = (ρσ)(mσ)(a) if equationis the unique expression of the element a of ∧ as a sum of elements ar of the∧r. Then {ф1 ,…,фc} is the complete set of inequivalent irreducible representations of Dσ . If dσ is the order of Dσ , then the degree of фσ is dσ.mσ. If F is algebraically closed, each Ds reduces to F and we may regard L as a direct sum of full transformation semigroup algebra ∧ over F. The irreducible representation of ∧ are then just the projections of τ upon its various components (see Th.7.3 in [4]).

THEOREM 7.4

Let τ be a linear operator on ∧ with an algebra ∧ of finite order over a field F.

If n > m, then there exists a non-zero linear transformation σ: ∧n →∧m such that τ σ = 0. There exists a non-null transformation γ : ∧n →∧m (over γτ ) such that γτ = 0, for every m > n.

Proof

Let n > m and equation with τ2 an operator on τ2 and τ2 a linear transformation from ∧n−m into ∧n−m (over τ1 ). Suppose that τ1 is left divisor of zero in equation(∧m) . then there existsequation such that τ1 σ1 = 0. We may take σ=(σ1,0). Hence we may assume that τ1 is not left divisor of zero in equation(∧m). By Lemma 5.8, that can be applied to the algebra equation(∧m) ), we have that the algebra τ1 contains a left identity element i with respect to which τ1 has a two-sided inverse ρ1 in τ1equation We may take equation where σ2 is any non-singular linear transformation from ∧m into ∧m over the algebra ∧ .

Then,

since equation and i is the identity element in equation(∧m).

One can similarly prove that, if m > n, then there exists a non-null transformation equation

Representation of a full transformation semigroup over a finite field

Let θ be a root of some irreducible polynomial of degree m over a finite field Fq(or the Galois field GF(q)), then the set {1, θ,θ2….,θm-1} becomes a basis for the vector space Fmq over Fq and is called a polynomial basis for Fmq. The dimension of the vector space Fmq over Fq is m. Letequation such that the set

equation

form a basis for equation So that a be represented by the vector (a0,a1,a2,…,am-1) and let αq be represented by the shifted vector (am-1,a0,a1,…,am-2). The normal basis exists for any extension field of Fq.

Consider the vector space V = Fqm over Fq (where q is a prime), and let equation= {θ,θq, θq ….,θqm-1} be a basis for V. Let TB be the full transformation semigroup upon the basis B. Then Tequation =mm.

Since equation is an element of V =Fqm as described above. Then the elementequation can be defined byequation then equation, where

equation

i.e., equation

It is obvious to say thatequation . S is a full transformation semigroup over V* with a dual basisequation of V* then there exists a mapping equation which becomes an isomorphism.

Since is a finite full transformation semigroup on the basis B of V over the finite field Fq. Therefore Fq [equation] becomes an algebra of equation over Fq. Then, there is a natural one-to-one correspondence between the representation of TB over Fq and those of equation which preserves equivalence, reduction and decomposition into irreducible constituents.

Thus the representations of equation over Fq is transferred to the algebra equation If equation is semisimple, then by the main representation theorem[4] holds for semisimple algebra equation Every representation of equation and hence every representation of TB is full reducible into irreducible one.

Let Fq be a finite field, and B be a basis for Fmq , where (m,q) = 1. (i.e., m,q are relatively prime).

Then, we have the following interpretation of the Maschke’s theorem regarding the algebra equation over the finite field Fq.

THEOREM 8.1

Let Let S =equation be a finite full transformation semigroup over basis equation of equation of order mm.

Then, the semigroup algebra equation over Fq is semisimple if and only if the characteristic q of Fq does not divides the order mm of the full transformation semigroup ∧.

Let ∧ be an algebra of order r over the vector space V = Fq m, and let n be another positive integer different from m. Denote by ∧ the full matrix algebra of all nn matrices over ∧ , with the additions and multiplication of matrices, and of the multiplication of matrix by a scalar in Fqm. Then, the algebra ∧ is of order rn2 over Fq m. In particular, (Fqm)n will denote the full matrix algebra of degree n over Fqm.

An algebra L over a field F is called division algebra if ∧ /0is a group under multiplication. A result regarding the existence of an isomorphism between a full matrix algebra and the space of all the linear transformations over the vector space Fqm , is as follows.

THEOREM 8.2

Let Fqm be a vector space over a finite field Fq. Then, there is an isomorphism from the space of full matrix algebra (Fq)m to the space equation of all the linear transformations on Fqm.

Proof

The set of all m−dimensional vector space (1m matrices) over Fq is an m−dimensional vector space Fqm over Fq . The natural basis of Fqm consists of the m vectors v1 = θ, v2 = θq, v3q2 ,…,vm = θqm-1, where vi has the identity element 1 of Fq for its ith component, and has 0 for the remaining components.

If A equation (Fq)m, then the transformation t : equation is a linear transformation t of Fqm into itself and the mapping equation is an isomorphism ofequation upon the algebraequation of all linear transformations of Fqm into itself. The ith row of A is the vectorequation

Conversely, if Fqm is any m−dimensional vector space, and we choose a basis {v1,v2,…,vm} of Fqm, then each linear transformation t of Fqm determines a matrix A = (αij) from the expression.

for the m vectors equation as linear combination of the basis vectors. Then, the mappingequation becomes an isomorphism of equation

Conclusion

A combinatorial result about the rank of a representation of the full transformation semigroup is obtained. It seems that for any homomorphism between the set of single-valued maps and the set of all nn matrices over a field F becomes a representation when the set of single valued maps is replaced by a full transformation semigroup adjoined with a zero element z. There is a one-one correspondence between the set of all representations of some finite semigroup S and those of the algebra of a full transformation semigroup over a finite dimensional vector space over a finite field. Consequently, we observed an isomorphism between the full matrix algebra (Fq)m and the set of all linear transformations on Fqm is obtained.

References