I nternat. J. Math. & Math. S ci.
487
Vol. 6 No. 3 (1983) 487-501
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
A. SITA RAMA MURTI
Department of Mathematics
Andhra University
Waltair 530 003, INDIA
(Received July 24, 1980 and in revised form April 17, 1981)
ABSTRACT.
If M is a centered operand over a semigroup S, the suboperands of M
containing zero are characterized in terms of S-homomorphisms of M.
Some properties
of centered operands over a semigroup with zero are studied.
A A-centralizer C of a set M and the semigroup S(C,&) of transformations of M
over C are introduced, where
&
is a subset of M.
irreducible centered operand over S(C,&).
semlgroups of transformations of sets
Mol
When
&
M, M
Theorems concerning the isomorphisms of
over
&.-centralizersl Ci,
obtained, and the following theorem in ring theory is deduced:
the rlngs of linear transformations of vector spaces
dimensional.
Then f is an isomorphism of L
i-I semilinear transformation h of M
KEY WORDS AND PHRASES.
I
L
I
onto M
2
2
(Mi,Di)
1,2 are
i
Let
Li,
i
1,2 be
not necessarily finite
if and only if there exists a
such that fT
hTh
-I
for all T
L
E
I.
Semigroups o f trans formations, operand over a semigroup.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
0.
is a faithful and
Primary 20M20, Secondary 20M30.
INTRODUCTION AND PRELIMINARIES.
In recent times Tully [I], Hoehnke [2], and others have studied the theory of
representations of a semigroup by transformations of a set.
This paper deals with
the study of a certain class of such representations (see Theorem 2.1).
In section
i we define an 0-suboperand of a centered operand M over a (general) semigroup and
characterize the same in terms of operand homoorphisms of M. Soe properties of
centered operands over semigroups with zero are discussed in Section 2. In Section
3 we introduce the concept of a &-centralizer C of a set M (with
non-empty subset
IMI
>
2), for any
& of M, and define the semigroup S(C,&) of transformations of
a
488
A. S. R. IIURTI
set M over C as the set of all self-maps of M which commute with every member of C.
We observe that M is a faithful centered operand over S(C,A), and also is irreducible in the case when A
M.
In Section 4 we obtain results (Theorems 4.1 and 4.2) which are comparable
[2], concerning the isomorphisms of semigroups of transforma-
with Theorem 17.3 of
Mo1 over centralizers
tions of sets
Ci,
for i
1,2, which generalize a similar
result concerning the isomorphisms of near-rings of transformations of groups (as
also analogous results for loop-near-rings)
Theorem 2.6 of Ramakotaiah [3]; then
we thereby deduce the following well-known isomorphism theorem in ring theory (see,
for instance, Jacobson [4]):
of vector spaces
(Mi, D i)
morphism of L
L
of M
I
onto
I
Let L i, i
1,2 be the rings of linear transformations
Then f is an iso-
not necessarily finite dimensional.
if and only if there exists a l-1 semilinear transformation h
2
M such that fT
2
hTh
-I
for all T
L
I.
Throughout this paper, by "an operand over a semigroup" we mean a left operand
only.
If M is a centered operand over a semigroup with zero, {0} and M are called
the trivial suboperands of M.
We often write 0 instead of (0}.
For the definitions
In Weinert [6],
and results on operands, we mostly follow Clifford and Preston [5].
the terms "S-set" and "S-mapping" are used to denote "operand over
S"
and "S-homo-
mo rphism" respectively.
The following definitions are taken from Santha Kumari [7].
A system N
(N, +,., 0) is called a loop-near-ring if the following condi-
tions are satisfied:
(i)
+
is a loop, which is denoted by N
(N,+,0)
(ii)
(N,.) is
(iii)
(a + b).c
(iv)
a.0
a semigroup
a.c
+
b.c for all a,b,c e N
0 for all a e N.
If N is a loop-near-ring, then an additive loop
provided there exists a mapping (n,g)
(i)
(ii)
(m + n)g
(mn)g
mg
+ ng
is called an N-loop
G, such that
ng of N x G
and
m(ng), for all m,n e N and g
(G, +, 0)
G.
489
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
i.
0-SUBOPERANDS OF A CENTERED OPERAND.
In this section, M denotes a centered (left) operand (see [5]) over a (general)
semigroup S and 0 denotes the fixed element in M.
homomorphism of M into a centered operand
DEFINITION.
M’,
is a S-
We observe that, if
then (0)
0.
A subset K of M is called an 0-suboperand of M if (if and only if)
S K __c K (that is, K is a suboperand of M) and 0 e K.
THEOREM 1.1.
Let M/K denote the
Suppose a subset K of M is a 0-suboperand of M.
Rees factor operand corresponding to the suboperand K of M and let
Clearly (x)
the canonical S-homomorphism.
For, if (t) is one such element, then (t)
S; in any case, we get that (t)
S with K as its zero and
-i
(K)
M/K be
Thus K
M/K
In fact, K is the only fixed element
s (t)
(st) for all s
and this gives that either t, st, or both belong to K for some s
all s
: M
K.
K if and only if x
and, moreover, K is a fixed element of M/K.
of M/K.
(0) for
of M.
some S-homomorphism
PROOF.
-i
A subset K of M is an 0-suboperand if and only if K
K.
S or t
st for
Hence M/K is a centered operand over
K.
The converse part can be easily proved by direct verification.
REMARKS 1.2.
Clearly {0} is the smallest 0-suboperand and M is the largest,
under set inclusion.
Also, the family F of all 0-suboperands of M is closed under
Hence, F is a complete lattice under set in-
arbitrary unions and intersections.
clusion, with set union and set intersection as the lattice operations.
It is a straightforward verification to see that
PROPOSITION 1.3.
S-homomorphism.
of
M’
Let
M’
be a centered operand over S and let
."
M
M’
be a
Then, (a) for every 0-suboperand K of M, (K) is a 0-suboperand
K’
and (b) for every 0-suboperand
PROPOSITION 1.4.
of
M’,
-I
(K’)
is a 0-suboperand of M.
Let K be a 0-suboperand of i and let
factor operand corresponding to K.
Let
:
M
M/K denote the Rees
M/K be the canonical homomorphism.
Then, (a) a subset B of M/K is a 0-suboperand of M/K if and only if
0-suboperand of M and (b) A
(A) is
a one-to-one
-i
(B) is a
correspondence between the sub-
operands of M containing K and the 0-suboperands of M/K.
PROOF.
S
(a) follows from Proposition 1.3, and the proof of (b) is routine.
A. S. R. MURTI
490
2.
ALMOST IRREDUCIBLE SUBOPERANDS AND ANNIHILATORS.
In this section we concentrate on centered operands over semigroups with zero,
and our study is motivated by the following:
THEOREM 2.1.
Let S be a semigroup with zero.
correspondence between the representations
9
Then, there exists a one-to-one
of S by transformations of a set such
that 9(0) is a constant map and the centered (left) operands over S.
PROOF.
9:
T
S
M
Let T
M
denote the full transformation semigroup of a set M and let
be a representation of S such that 9(0) is a constant map.
9(a)(x) for all
operand over S with multiplication defined by a.x
Let 9(0)(M)
9(a0)(t)
{t}.
For any a
9(0)(t)
9:
T
S
9(a) (9(0) (t))
9(a)(t)
t and so t is a fixed element of M.
is a fixed element of
operand over S.
S, a.t
M, then we have y
9(0)(y)
t.
a
Now M is an
M.
S, x
(9(a)9(0))(t)
On the other hand, if y
Hence M is a centered
Conversely, if M is a centered operand over S, then the map
by 9(a)(x)
M given
transformations of M such that
a.x for all a e S, x
9(0)(x)
M is a representation of S by
0 for all x e M.
Hence the result.
Throughout the rest of this section, S denotes a semigroup with zero and
M # 0 denotes a centered (left) operand over S.
For any centered operand N over S
and a suboperand K of N, N/K denotes the Rees factor operand corresponding to K.
DEFINITION.
M is said to be almost irreducible (a. irreducible) if M has no
nont rivial suboperands.
Clearly, irreducibility (see [5]) implies a.irreducibility.
REMARKS 2.2.
Also, a. irreducibility implies irreducibility except possibly in the case when M
has exactly two elements (also see Proposition 2.4 below).
genic’ synonymous to ’strictly cyclic’.
M.
M is said to be strongly monogenic if to each t
M, St
We note that M can be strongly monogenic without being monogenic.
presence of SM # 0,
’M
’mono-
We say that M is monogenic by t (or, equi-
valently, t is an S-generator of M) if and only if St
DEFINITION.
We use the term
is strongly monogenic’ implies ’i is monogenic’.
0 or M.
But in the
The fol-
lowing results are easy consequences of the above definitions.
PROPOSITION 2.3.
then K
M.
If K is a suboperand of M and k
K is an S-generator of M,
491
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
PROPOSITION 2.4.
M is irreducible if and only if M is a.irreducible and mono-
genic.
DEFINITION.
M is said to be faithful if the representation associated with M
is faithful (see [i]).
DEFINITION.
Let C be a nonempty subset of M.
Then {s e S
sC
0} is called
For any t e M, A({t}) is denoted by
the annihilator of C and is denoted by A(C).
A(t).
PROPOSITION 2.5.
For any nonempty subset C of M, A(C) is a left ideal of S.
In particular, A(t) is a left ideal of S for each t e M.
PROOF.
PROPOSITION 2.6.
PROOF.
s
A(C).
It can be directly verified that S.A(C)
If M is faithful, then A(M)
Let s e A(M).
Then, for t
0.
M we have st
0
0t and this gives
0 since M is faithful.
PROPOSITION 2.7.
Suppose M is a. irreducible.
Then the following hold.
(a)
M is strongly monogenic
(b)
If L is a left ideal of S, then, for any t
(c)
If A(M)
(a) is obvious, since Sx
L # O, it follows that L
M
M.
is a suboperand for each x e M.
if we observe that Lt is a suboperand of M.
Hence, Lt
0 or M.
0 and 0 # L is a left ideal of S, then there exists t
such that Lt
PROOF.
M, Lt
A(M).
Now we prove (c).
(b) is clear
Since A(M)
0 and
Therefore, there exists t e M such that Lt # O;
M.
__
PROPOSITION 2.8. Let 0 # L be a left ideal of S. If L is a. irreducible as an
operand over S (in the natural way), then L is a O-minimal left ideal of S.
PROOF.
Let J be a left ideal of S with 0
the operand L over S.
J
L.
Then J is a suboperand of
Since L is aoirreducible, we have J
0 or L.
Hence the
result.
DEFINITION.
-i
(0)
M is said to be smooth if any S-homomorphism
of M satisfying
0 is injective.
DEFINITION.
If
called the kernel of
.
is an S-homomorphism of M, then the
and is denoted by ker
congruence
-i
o
is
492
A.S.R. MURTI
PROPOSI.TION 2.9.
The following are equivalent:
(a)
M is a primitive operand (see [i]) over S.
(b)
For any S-homomorphism
(the diagonal of M x M) or
of M, ker
MxM.
(c)
M is smooth and a. irreducible.
0
with
-1
(b) is trivial.
(a)
PROOF.
(0)
0.
Assume (b).
Therefore, ker
and so
Let
be an S-homomorphism of M
To show that m is a. irreducible, let K be a suboperand of M.
of M.
some S-homomorphism
Thus (c) is proved.
Hence M is smooth.
is injective.
-i(0) for
-i(0) 0 or M.
Then K
But from hypothesis, if follows that
To prove (a), it is enough to prove (b),
Finally, assume (c).
since every congruence in M is the kernel of some S-homorphism of M.
be an S-homomorphism of M.
hence
Then
-I
0 or M (since M is a.irreducible) and
(0)
is the zero map.
is injective or
Now, let
or M x
Therefore, ker
M, proving
(b).
-i
Let K
epimorphism.
M,M’
Let
THEOREM 2.10.
M’
If M/K is smooth over S, then
(0).
@: M
Let
be centered operands over S.
/
M’
be an S-
is S-isomorphic to
M/K.
PROOF.
Let :
get that ker
.
M- M/K be the canonical homomorphism.
ker
is the zero of
(x) for all x
Therefore, "h((x))
epimorphism h of M/K onto M’.
Since K
Further, h((x))
e
M"
-i(0),_
we
define an S-
0 if and only if (x)
K, which
M/K, and, since M/K is smooth, it follows that h is injective.
Thus
h is an isomorphism.
THEOREM 2.11.
Suppose M is irreducible.
For any non-zero t
M, if S/A(t)
is smooth over S, then A(t) is a maximal left ideal of S.
PROOF.
Let 0 # t
t is an
S-generator of M,
the map
t:
s
M and assume that S/A(t) is smooth.
by Lemma II.16(B) of [5].
Since M is irreducible,
Therefore A(t) # S.
st from S into M is an S-epimorphism, and
-i(0)
A(t).
Also,
Now, by
Theorem 2.10, S/A(t) is isomorphic to M and therefore S/A(t) is a. irreducible.
A(t) ! L is a left ideal of S, then by Proposition 1.4 it follows that (L) is a
suboperand of S/A(t) where :
then, (L)
S
S/A(t)
is the canonical S-homomorphism.
A(t) or S/A(t) which gives that L
A(t) or S.
But
Hence the result.
If
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
Suppose M is a. irreducible.
THEOREM 2.12.
M,
(0)
0 implies
PROOF.
Since L
Moreover,
hence is 0.
DEFINITION.
M such that Im # O.
A(C) there exists m
Therefore the map
@-i(0)
Therefore
Then L is S-isomorphic to M.
is injective.
by Proposition 2.7(b).
phism.
Let L be a 0-minimal left ideal of
A(C) for some C c__ M and (ii) for any S-somomorphism e of L into
S such that (i) A
-i
493
@
@:
%
Therefore Lm
M
%m from L onto M is an S-epimor-
is a left ideal of S and is properly contained in L, and
is injective, by (ii) of hypothesis.
Hence the result.
S is said to be primitive if S admits a faithful and irreducible
centered operand.
If M is one such operand, we say that S acts primitively on M.
Now, Theorem 2.12 yields the following, by taking M for C.
COROLLARY 2.13.
Let S act primitively on M and let L be a 0-minimal left
ideal of S such that, for any S-homomorphism e of L into M,
is injective.
3.
-i
(0)
0 implies
Then L is S-isomorphic to M.
SEMIGROUPS OF TRANSFORMATIONS OVER A CENTRALIZER.
Here we mainly introduce two concepts, namely (i) a centralizer C of a non-
empty set M in a generalized form and (2) the semigroup S(C) of transformations
(of M) over a centralizer C of M, and study some preliminary properties of the
centered operand M over S(C).
Theorem 3.7 plays the key role in deducing the
corresponding results for near-rings, of [3], and loop-near-rings from some of our
main results.
2 and such that 0 e M is
Throughout this section, M denotes a set with IMI
I denotes the identity mapping on M and 0, the constant
a distinguished element.
map on M with range {0}.
By an endomorphism of M, we mean a mapping of M into itself fix-
DEFINITION.
ing 0.
A bijective endomorphism of M is called an automorphism of M.
DEFINITION.
Let
be a non-empty subset of M.
A set C of endomorphisms of M
is called a A-centralizer of M if
(i)
(ii)
(iii)
(iv)
0
c
C- 0 is a group of automorphisms of M
e(A) c__ A for all e
,
C, 0 # w
C- 0
A and
(w)
B(w) imply
494
A. S. R. MURTI
If A
M, then a A-centralizer of M is referred
The set
{I,0}
,
is a A-centralizer of M for any A
{0,a,b,c}
example, take M
to as a centralizer of
{I,
and let C
keeping the other elements fixed.
M.
} where
Mo
To get a non-trivial
interchanges a and b
Then C is a A-centralizer of M where A
{a,b}.
Evidently, any centralizer of a group G (see Ramakotaiah [8], Definition 2)
is a centralizer of the set G (with the identity element of G acting as the distin-
guished element).
We notice that M is a vector set in the sense of [2], over any
centralizer of M.
LEMMA 3.1
Let C be a set of endomorphisms of M containing 0 such that C
is a group of automorphisms of M.
Then C is a A-centralizer of M for some subset
of M containing non-zero elements of M if and only if
PROOF.
# M.
Suppose M
I
{x e M
Write F
Put A
M
M
M
I.
Then
A
-1
Hence 8(A)
CA.
and
$-i
(w)
{x e M
u
C-O
e C
and put
(x)=x} # M.
u
F
c
C-0
M
I
contains a non-zero element, and we shall
Let w
which implies that there exists
1
Now I # 8
x} for each
(x)
show that C is a A-centralizer of M.
8(w)
0
A
and
C-0,
e
C-O, with (w)
A.
# I such that ((w))
w which says that w
A,
Then
(w).
a contradiction.
The rest is also similar.
Conversely, if C is a A-centralizer of M such that A contains a non-zero eleand the proof
hence M #
ment say w, then it can be easily verified that w
MI;
MI,
is complete.
In the rest of this section, C denotes a non-trivial A-centralizer of M with
0
Ta
A.
DEFINITION.
A mapping T of M into M is called a transformation of M over C if
sT for all
e C.
REMARK 3.2.
Any transformation of M over C fixes O.
mations of M over C, denoted by
S(C,A), is
The set of all transfor-
a semigroup with zero and unity element
(under composition of mappings) and M is a centered operand over S(C,A) in a natural
way.
Moreover M is faithful.
a straightforward
In case A
M, we shall denote S(C,A) by S(C).
verification, one can see that:
By
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
For any
PROPOSITION 3.3.
e C,
{x
in A defined by x
The relation
x} is
e(x)
M
495
a suboperand of M.
e C
y if and only if there exists
0
y is clearly an equivalence relation on A and the equivalence
such that (x)
classes are called the orbits of C on
A.
The following lemma can be proved on the
same lines as in Lemma 8 of [8] and is a generalization of the latter.
Let 0 # w
LEMMA 3.4.
A and w’
M.
w’ and (ii) T maps elements of M
(i) T(w)
Then there exists T
S(C,A) such that
which do not belong to the orbit of w
onto 0.
It follows from Lemma 3.4 that every non-zero element of A is a
REMARK 3.5.
S(C,A)-generator of M.
Hence, if C is a centralizer of M, then S(C) acts primitive-
If M is a group (respectively loop) and A is a non-empty subset of M,
ly on M.
then we can analogously define (i) a A-centralize# C of the group (loop) M- so
that it reduces to the centralizer of the group (loop) M when A
M- and (2) the
near-ring N(C,A) (loop-near-ring L(C,A)) of all transformations of M over C.
M is a faithful N(C,A)-group (L(C,A)-loop).
coincide with our S(C,A).
COROLLARY 3.6.
Then
Also, as sets, N(C,A) and L(C,A) both
Thus we have:
Let M be a group (loop), and 0 # A, a subset of M containing
Then every non-zero element of A
0 and C, a A-centralizer of the group (loop) M.
M, then M is a N(C)N(C,A)-generator (L(C,A)-generator) of M. Hence, if A
group (L(C)-loop) of type 2 and N(C) (L(C)) acts 2-primitively on M.
is a
Using Lemma 3.4, we obtain the following theorem which is crucial in extending
some of our main results to near-rings (loop-near-rings) of transformations of a
group (loop) M over a centralizer of the group (loop) M.
THEOREM 3.7.
(L(C,A)-loop).
Let M,C be as in Corollary 3.6.
Let M’ be a N(C,A)-group
Then any S(C,A)-(operand) homomorphism
9
of M into
M’
is a
N(C,A)-
group (L(C,A)-loop)homomorphism (that is, preserves addition also); hence, if
9-1(0)
0, then 9 is injective.
PROOF.
A.
Let x,y
Tl(W)
(w)
x,
T
1
Let 9"
M.
T2(w)
M
M’
be an S(C,A)-homomorphism.
Then, by Lemma 3.4, there exist
y.
9(w) + r 2 9(w)
Now 9(x
+ y)
9 T l(w) + 9
TI,T 2
Fix a non-zero element w of
N(C,A)(=S(C,A)) such that
e
9(TI(W)
+
r2(w)
9(x) + 9(Y).
T2(w))
9(T
I
+ T 2)(w)
(T
I +
Hence the result.
T2)9
496
A. S. R. MURTI
_
Theorem 3.7 can be generalized to the case of Universal Algebras, as follows.
We assume that (A,) is a Universal algebra such that A has a distinguished element
0 (that is, some f
DEFINITION.
izer of the
A.
A set C of endomorphisms of the -algebra A is called a A-central-
-algebra A if (i) 0
A for all
(iii) (A)
A
is nullary) and 0
0 is a group of automorphlsms of A
C (ii) C
C and (iv)
,
C, 0 # w
A, (w)
all transformations of A which commute with every member of C.
Universal algebra
(U(C,A),
{o}). Now A
u
.
We denote by U(C,A), the set of
Let C be a A-centralizer of the -algebra A.
pointwise, and adding the binary operation
(w) imply
Defining operations
"o" of composition of mappings,
is a centered operand over
we get a
U(C,A) and we
have the following theorem whose proof is similar to that of Theorem 3.7.
Let B be a -algebra such that there
Let A, U(C,A) be as above.
THEOREM 3.8.
(left) multiplication of the elements of B by the elements of U(C,A), satisfy-
is a
ing (i) f(T I,
and b
B (ii)
Tn).b f(Tl’b,
(TIT2).b TI’(T2"b)
,
for
Tn’b) for all f
all TI, T
U(C,A)
2
TI,
T
B.
and b
U(C,A)
n
Then any
U(C,A)-(operand) homomorphism of A into B is a -algebra homomorphism.
With the usual notation, we have:
LEMMA 3.9.
the orbit of w.
PROOF.
That
A.
Let 0 # w
Then M is S(C,A)-isomorphlc to A(M-) where
Consider the S(C,A)-homomorphism
T
is surjective follows from Lemma 3.4 and
using the definition of S(C,A).
THEOREM 3.10.
orbit.
/
T(w) from A(M-) into M.
can be shown to be injectlve
Hence the result.
Suppose M is a.irreducible over S(C,A).
Let
be a non-zero
Then A(M-) is an irreducible operand over S(C,A) and hence
minimal left ideal of S(C,A); further,
PROOF.
A(M-F)
S(C,A) and thereby T
M
M which is identity on
A(M-P)
a 0-
To prove the last
and 0 elsewhere.
Now
A(A).
The bracketed statement of the following corollary is due to
COROLLARY 3.11.
A(M-) is
A(A).
The first part is an easy consequence of Lemma 3.9.
part, consider the map T:
T
:
is
[3], Lemma 2.1.
Let M # 0 be a group (loop) and C, a centralizer of M.
Then
N(C)(L(C)) contains a left ideal K which is N(C)-group isomorphic (L(C)-loop iso-
497
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
morphic) to M and hence is a N(C)-group (L(C)-loop) of type 2.
Afortiori, K is a
minimal left ideal of N(C)(L(C)).
C is a centralizer of also the set M and N(C), L(C) are both equal to
PROOF.
S(C), as sets.
A simple verification shows
be a non-zero orbit of C on M.
Let
that A(M-P) is a left ideal of the nar-ring (loop-near-ring) N(C).
By Lemma 3.9,
M is S(C)-operand isomorphic to A(M-) and so by Theorem 3.7, M is N(C)-group isomorphic (L(C)-loop isomorphic) to A(M-r).
Since M is a N(C)-group (L(C)-loop) of
type 2 (see Corollary 3.6), so is A(M-).
The rest follows from Theorem 3.10.
As an immediate consequence of Corollary 2.13 we have:
Suppose M is a.irreducible and let L be an 0-minimal left
PROPOSITION 3.12.
ideal of S(C,A) suuh nat for any S(C,A)-homomorphism
Let C’ be a A-centralizer of M such that C c__ C’.
THEOREM 3.13.
S(C’,A) if and only
PROOF.
that
T
so
# C’.
0
Then S(C,A)=
S(C’,A) and
To prove the converse, let S(C,A)
Then there exists
r,
e’ (w)
(0)
C’.
if C
One way is clear.
assume that C
-I
Then L is S (C A) -isomorphic to M.
is injective.
implies
of L into M, a
’
C’
C.
Let 0 # w
A.
It can be seen
the orbit of w with respect to C. Now by Lemma 3.4, there exists
S(C,A) such that T(w)
a’(w) and T maps M-[ onto 0. But then T S(C’,A) and
a’T(w)
Ta’(w)
0.
COROLLARY 3.14.
of M.
Therefore w
([3],
N(C’)
Then N(C)
0, a contradiction.
Theorem 1.2.) Let M be a group and C &
if and only if C
C’,
centralizers
C’.
The following result generalizes Corollary 1.3 of
[3] (as also the analogous
result for loops) and the proof is analogous to that of the latter.
PROPOSITION 3.15.
phism
of M,
-I
a
(0)
0 implies e is injective and (c) A-0 is the set of all
S(C,A)-generators of M.
aT
Suppose (a) M is a.irreducible (b) for any S(C,A)-endomor-
Then the set of all endomorphisms of M satisfying (i)
T for all T e S(C,A) and (ii) a(A) c__
PROPOSITION 3.16.
A,
is C itself.
If A’ is the set of all S(C,A)-generators of M together
with zero, then C is a A’-centralizer of M and S(C,A)
PROOF.
e(M)
M.
Let e
C.
Then for any 0
# w
Therefore, e(A’) m A’ for all a
S(C,A’).
A’, S(C,A)(a(w))
a(S(C,A)(w))
C and, similarly, the other conditions
can be verified to show that C is A’-centralizer of M.
The rest is obvious.
498
A. S. R. MURTI
In Proposition 3.16, there is no harm in taking M as a group (or a loop) and
C as a A-centralizer of the group M (loop M).
4.
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS.
_
We introduce here the concept of a generalized semi-space as a generalization
of semi-space introduced by [3].
DEFINITION.
M, 0
with 0
A generalized semi-space is a triple (M,A,C) where M is a set
A
M and C is a A-centralizer of M.
If A
M,
we omit A and write
simply as (M,C).
Let (Mi, A i,
DEFINITION.
:
C
-
l
C
2
be generalized semi-spaces for i
Ci)
is called an isomorphism of C
isomorphism of C
0 onto C
1
if (0)
I onto C 2
0 and
denotes a generalized semi-space for i
into M
C
I
2
A map h:
M
if (i) h fixes 0 and
onto C
2
such that ha
by (h,).
I
M
h(Al)
We notice that, if
is called a semi-linear transformation of M
2
c__
42
and (ii) there exists an isomorphism
C
(GI,C I)
and
formation of the generalized semi-spaces
I.
M
A semilinear transformation h:
I)
A
M
I
2
is called a i-i semilin-
2.
If (h,) is a i-i semilinear transformation of M I onto M 2 (h
2
onto M
I.
of
(G2,C 2) are semi-spaces, then any semi(GI,C I) and (G2,C 2) is a semi-linear trans(GI,C I) and (G2,C2).
ear transformation if h is bijective and h(A
such from
I
also, we shall denote the semilinear transformation
linear transformation of the semi-spaces
DEFINITION.
(Mi’ Ai’ Ci)
1,2.
(a)h for all
If we wish to indicate
is a group
0.
2
Throughout the rest of this paper, unless otherwise stated,
DEFINITION.
A map
1,2.
-I, G -1)
is one
The proof of the following theorem is analogous to that of
Lemma 2.7 of [3].
Let (h,) be a i-i semilinear transformation of M I onto M 2.
-I for all T
onto
defines an isomorphism of S(C I, A
S(C I, A
hTh
THEOREM 4.1.
Then, (T)
S (C 2,
I)
I)
A2).
Conversely,
THEOREM 4.2.
Let
be an isomorphism of S(C I, A
that Mo is a. irreducible over S(C i, A i) for i
1
faithful, irreducible operand over S(C I, A
I).
1,2.
I)
onto
S(C 2, A 2) and suppose
Then M2 can be regarded as a
Further, suppose that
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
(1)
foe any
S(CI, Al)-homomorphism
e of M
I
-I
M2,
into
499
(0)
0 implies
is inj ec t ive.
(ll)
(ill)
for i
1,2,
for any
S(CI, Ai)-endomorphlsm
Ai-0
S(Ci, Ai)-generators of M i-I
0 implies e is
(0)
Mi,
is the set of all
Injectlve, for i
of
1,2.
Then there exists a i-i semillnear transformation (h,) of M
is an
S(CI, Al)-Isomorphism
of M
1
onto M
2
I onto M2 such that h
hTh-lfor
and (T)
all T
S(C I,
AI).
Before proving this theorem, we give the following three lemmas in each of
Which it is assumed that
1,2, Mi is a.lrreduclble over
for i
LEMMA 4.3.
S(C 1,
and m
is an isomorphism of
c
M2 c=-
S(CI, A I)
onto
S(C2, A 2)
and that
S(CI,
regarded as a faithful and irreducible operand over
41)PROOF. The left multiplication ’.’ given by T’m
M2 serves the purpose.
Then there exists an
hTh
-I for all
PROOF.
S(CI,4 I)
Suppose conditions (i) and (il) of Theorem 4.2 are also satisfied.
LEMMA 4.4.
@(T)
(T)(m) for each T
r
Let
S(CI, 41)-isomorphism
T
S(CI! 41).
be a non-zero orbit of C
S(CI, 41)-Isomorphlc
to
A(MI-).
h of M
I
onto M
I
42
I)
and
Lemma 3.9 says that MI is
41
over
such that h(4
2
Using Theorem 3.10 and condition (1) of the hy-
pothesis, we get from Proposition 3.12 that
A(MI-[)
is S(C I,
41)-isomorphlc
to M
2.
S(CI, 41)-isomorphism h: MI M2. Now, let T S(C I, 41 and
@(T)h, which means
@(T) h(m I) and hence hT
T’h(m I)
Then h T(m I)
m
I MI.
-I
It remalns to show that h(4 I)
(r)
hrh
42 Let 0 # wI 41 Then
S(C2, 42) h(Wl) (S(CI, 41))h(w I) (h S(C I, 41)h-l)h(w I) h S(C I, 41)(w I)
h(M I)
M2. Therefore, h(w I)
42 by condition (ii). Thus h(4I) 42 To prove
the reverse inclusion, let 0 # w2
42 Then w2 is an S(CI, 41)-generator of M2,
h-i (w2) h -i (S(C I, 41)w2) h-l(M2 MI. Therefore, h-I(42)c_I
and so S(C
I 41
So, there exists an
/
and this completes the proof.
isomorphism of M
I
onto M
2
such that h(4
I)
42
CI
and
:
I
+
hlh-i
-I for all T
hTh
and (T)
(the existence of h being ensured by Lemma 4.4).
el
41)S(CI,4 I)
Assume all the hypothesis of Theorem 4.2, and let h be an S(C I,
LEMMA 4.5.
is an isomorphism of C
Then h
I
el
onto C
2.
h
-i
C
2
e
for each
A. S. R. MURTI
500
PROOF.
Let
i
e
CI"
_
.
Write
suffices to show that (a)
hlh
2
-i
In view of Proposition 3.15, it
is an endomorphism of M
2
2
and (b)
2(A2)
n A
2
and (c)
S(C 2,
A2).
A and h is an isomorphism, we have 2(A2)
(a) is obvious. Since h(A I)
2
(b). Finally, let T 2
proving
S(C2, A 2) and
hlh-l(A2) hl(Al) h(Al) A2,
T and
S(C I, A I) and m E M1 such that (T I)
Then there exist T
m
2
I
1
2
h(ml) m2. Now 2T2(m2 hlh-l(Tl)h(m I) hlh-Tlh-(m I) hiTl(ml)
hTll(ml) hTlh-lhlh-lh(ml T22(m2), which proves (c).
for all T
2
2T2 T22
In view of Lemmas 4.3, 4.4 and 4.5, it remains to show
PROOF OF THEOREM 4.2.
that
O(l)h,
implies ha I
C1
i
which is clear from the definition of
o.
Hence
the theo rem.
The particular case of Theorem 4.2 when A
REMARK.
M can also be deduced
from [2], Theorem 17.3.
COROLLARY 4.6.
2.6 of [3]).
Let
(Isomorphism Theorem for Near-rings of Transformations, Theorem
(Gi, Ci)
(GI,
1,2, be semi-spaces
i
there exists a I-i semi-linear transformation h of G
morphism of
N(CI)
N(CI)
PROOF.
(a):
for all A
defines an isomorphism of N(C
I)
1
onto
G
are groups).
2
G2,
onto
N(C2).
(a) If
then f(A)
hAh
-i
(b) If f is an iso-
N(C 2), then there exists a i-I semi-linear transformation h
-I
for all A E N(CI).
of G onto G such that f(A)
hAh
2
1
semi-spaces
A
N(CI)
onto
Clearly h is a i-i semilinear transformation of the generalized
(GI, CI)
(G2, C2).
and
Now by Theorem 4.1, f(A)= bah
Let
show that f preserves addition also.
f(A +
hA2h-l(g2)
f(Al)
h(A +
A2)(g2)
1
+
1
f(Al)(g2)
f(A2).
(b):
N(CI)
defines a multiplicative semigroup isomorphism of
A
2
e
N(CI)
and
h(Alh-l(g2) + A2h-l(g2)
A2)h-l(g2
+
AI,
f(A2)(g2)
(f(Al)
+
f(A2))(g2)
g2
-i
for all
onto
N(C2).
G2"
We have
hAlh-l(g2
hence, f(A 1 +
We
+
A2)
Thus f is a near-ring isbmorphism.
From Lemma 3.4 we get that every
We deduce this part from Theorem 4.2.
non-zero element of G. is an
a. irreducible operand over
satisfied here.
from Theorem 3.7.
N(Ci)-generator
N(Ci)
for i
of G
i
for i
1,2 and so G. is an
1,2 and condition (ii) of Theorem 4.2 is
That conditions (i) and (iii) are also satisfied here, follows
Hence there exists a I-i semilinear transformation h of the gen-
501
ISOMORPHISMS OF SEMIGROUPS OF TRANSFORMATIONS
eralized semi-space (G I, C
onto G
2
hAh
and f(A)
-I
I)
(G2, C such
onto
N(CI)
for all A
that h is an
N(Cl)-isomorphism
I
By Theorem 3.7, h is a group isomor-
phlsm too, and hence h is a i-i semilinear transformation of the semispaces
(GI,C I)
Hence the result.
(G2, C2).
and
of G
We now get the following Isomorphism Theorem for rings of linear transformations
of vector spaces over division rings, the proof being the same as that given in [3],
Corollary 2.13.
Let
COROLLARY 4.7.
vector spaces
of L
I
2
I
-
(Mi, D i)
Li,
i
1,2, be the rings of linear transformations of
not necessarily finite dimensional.
Then f is an isomorphism
L 2 if and only if there exists a I-i semilinear transformation h of M I onto
such that fT
REMARK 4.8.
hTh
-i
for all T
L
I.
In the case of loops also, we can define semi-spaces and their
semilinear transformations analogously, and all the corollaries obtained in this
section for groups hold for loops as well, with
replaced by
’near-ring of transformations’
loop-near-ring of transformations’.
ACKNOWLEDGEMENTS.
I thank Dr. N.V. Subrahmanyam for his valuable suggestions
during this work.
REFERENCES
I.
TULLY, E.J., Jr.
2.
HOEHNKE, H.J.
3.
RAMAKOTAIAH, D. Isomorphisms of near-rings of transformations, J. Lond. Math.
Soc. (2), 9 (1974) 272-278.
4.
JACOBSON, N.
5.
CLIFFORD, A.H. and G.B. PRESTON.
Representation of a semigroup by transformations acting
transitively on a set, Amer. J. Math. 83 (1961) 533-541.
Surveys
Structure of semigroups, Canad. J. Math. 18 (1966) 449-491.
Structure of rings, A.M.S. Colloquium Publ., Vol. XXXVII, (1956).
The algebraic theory of semigroups, Math.
of the American Math. Soc. (1967).
(Providence, RI, Vol. 2).
6.
WEINERT, H.J. S-sets and semigroups of quotients, Semigroup Forum, Vol. 19,
(1980) 1-78.
7.
KDR,ARI, C. SANTHA.
8.
RAMAKOTAIAH, D.
Theory of loop-near-rings, Doctoral Thesis, Nagarjuna
University, India (1978).
Structure of 1-Primitive near-rings, Math. Z. ii0 (1969) 15-26.