Journal of Mathematical Extension
Vol. 9, No. 3, (2015), 97-108
ISSN: 1735-8299
Journal
of Mathematical
Extension
Journal
of Mathematical
Extension
URL:
http://www.ijmex.com
9, No.
3, (2015),
97-108
Vol.Vol.
9, No.
3, (2015),
97-108
ISSN:
1735-8299
ISSN:
1735-8299
URL:
http://www.ijmex.com
URL:
http://www.ijmex.com
A Study of Decomposer and Related
Functions on Groups
Study
Decomposer
and
Related
AA
Study
of of
Decomposer
and
Related
M. H. Hooshmand
Functions
on
Groups
Functions on Groups
Shiraz Branch, Islamic Azad University
Hooshmand
M.M.
H. H.
Hooshmand
Shiraz
Branch,
Islamic
Azad
University
Shiraz
Branch,
Islamic
Azad
University
1.
Abstract. Decomposer functions in algebraic structures are studied
in many recent papers. They have close relations to factorization by two
subsets. Also, idempotent endomorphisms form a class of (strong) deAbstract.
Decomposer
functions
inalgebraic
algebraic
structures
studied
Abstract.
Decomposer
functions
algebraic
structures
are
studied
composer
functions
in groups.
Now, ifinthe
structure
is aare
group,
in
many
recent
papers.
They
have
close
relations
to
factorization
by two
in
many
recent
papers.
They
have
close
relations
to
factorization
by
two
then by introducing a type of local homomorphisms we obtain several
subsets.
Also,
idempotent
endomorphisms
form
a class
of (strong)
subsets.
Also,
idempotent
endomorphisms
form
a class
of
de- deproperties
and
equivalent
conditions
for many
classes
of (strong)
decomposer
composer
functions
in
groups.
Now,
if
the
algebraic
structure
is
a
composer
functions
in
groups.
Now,
if
the
algebraic
structure
is
a
group,
functions and get a new result regarding to factorization of a group group,
by
then
by introducing
a we
type
of local
homomorphisms
obtain
several
then
bysubsets.
introducing
a type
ofprove
local
homomorphisms
we we
obtain
several
its two
Moreover,
existence
of (two-sided)
decomposer
equivalent
conditions
many
classes
of decomposer
properties
andand
conditions
for for
many
classes
of decomposer
typeproperties
functions
inequivalent
non-simple
groups.
functions
a new
result
regarding
to factorization
a group
functions
andand
get get
a new
result
regarding
to factorization
of aofgroup
by by
its
two
subsets.
Moreover,
we
prove
existence
of
(two-sided)
decomposer
its
two
subsets.
Moreover,
we
prove
existence
of
(two-sided)
decomposer
AMS Subject Classification: 20F99; 39B52
type
functions
in non-simple
groups.
type
functions
inPhrases:
non-simple
groups.
Keywords
and
Functional
equation, decomposer function, local homomorphism, factorization by subsets
AMS
Subject
Classification:
20F99;
39B52
AMS
Subject
Classification:
20F99;
39B52
Keywords
Phrases:
Functional
equation,
decomposer
function,
Keywords
andand
Phrases:
Functional
equation,
decomposer
function,
lo- lohomomorphism,
factorization
by subsets
cal cal
homomorphism,
factorization
by subsets
Introduction
There are many classes of functions from algebraic structures to themselves
1. have
Introduction
1.
Introduction
which
close relations to some aspects of algebraic properties. One of them
is the class of decomposer type functions, introduced and studied in [2]. They
There
many
classes
of functions
from
algebraic
structures
to themselves
There
are are
many
classes
of functions
from
algebraic
structures
to themselves
have many relations to factorization by two subsets, associative , multiplicative
which
have
close
relations
to some
aspects
of algebraic
properties.
of them
which
have
close
relations
to some
aspects
of algebraic
properties.
OneOne
of them
symmetric and canceller functions (see [1,4,6]). Indeed, decomposer type funcis the
class
of decomposer
type
functions,
introduced
studied
in [2].
They
is the
class
of decomposer
type
functions,
introduced
andand
studied
in [2].
They
tions satisfy some functional equations on algebraic structures (see [5,7]). In the
have
many
relations
to factorization
by two
subsets,
associative
, multiplicative
have
many
relations
to factorization
by two
subsets,
associative
, multiplicative
case that the algebraic structure is a group, the properties are so much more and
symmetric
canceller
functions
[1,4,6]).
Indeed,
decomposer
type
funcsymmetric
andand
canceller
functions
(see(see
[1,4,6]).
Indeed,
decomposer
type
functhere are many important sub-classes. Fore instance, cyclic decomposer (parter)
tions
satisfy
some
functional
equations
on algebraic
structures
[5,7]).
In the
tions
satisfy
some
functional
equations
on algebraic
structures
(see(see
[5,7]).
In the
functions that are periodic, and specially b-parts functions on the additive real
algebraic
structure
a group,
properties
so much
more
casecase
thatthat
thethe
algebraic
structure
is aisgroup,
thethe
properties
are are
so much
more
andand
numbers group (see [3]). We found that they have also some connections to a
there
many
important
sub-classes.
Fore
instance,
cyclic
decomposer
(parter)
there
are are
many
important
sub-classes.
Fore
instance,
cyclic
decomposer
(parter)
b-parts
functions
on the
additive
functions
that
periodic,
specially
b-parts
functions
on the
additive
realreal
functions
that
are are
periodic,
andand
specially
Received:
December
2014; Accepted:
May 2015
numbers
group
found
they
have
some
connections
numbers
group
(see(see
[3]).[3]).
WeWe
found
thatthat
they
have
alsoalso
some
connections
to ato a
97
Received:
December
2014;
Accepted:
Received:
December
2014;
Accepted:
MayMay
20152015
97 97
98
M. H. HOOSHMAND
type of local homomorphisms which we call them ∆ × Ω-homomorphisms. Let
us recall decomposer type functions on groups (from [2,4]) first, and then introduce the new conceptions. By applying them, we obtain many new results
regarding to decomposer type functions on groups and prove some related theorems.
Let (G, .) be a group with the identity element e. If f and g are functions from
G to G, then define the functions f.g and f − by
f.g(x) = f (x)g(x) ,
f − (x) = f (x)−1 :
∀x ∈ G.
We denote the identity function on G by ιG and put f ∗ = ιG .f − , f∗ = f − .ιG
and call f ∗ [resp. f∗ ] left ∗-conjugate of f [resp. right ∗-conjugate of f ]. They
are also called ∗-conjugates of f . Note that f ∗ (e) = f∗ (e) = f − (e) = f (e)−1 .
If (G, +) is additive group, then the notations e, f − , f.g, f.g − are replaced by
0, −f , f + g, f − g and we have f ∗ = f∗ = ιG − f . It is easy to see that
f is idempotent ⇔ f ∗ f = e ⇔ f∗ f = e
2
f ∗ = f ∗ ⇔ f f ∗ = e ⇔ f ∗∗ f ∗ = e , f∗2 = f∗ ⇔ f f∗ = e ⇔ f∗∗ f∗ = e.
Also, if f is endomorphism then f − f = f f − , f ∗ f = f f ∗ and f∗ f = f f∗
(i.e. the compositions of f and its ∗-conjugations are commutative).
Now, we recall from [2,4] decomposer type functions as follows.
A function f from (a group) G to G is called:
(a) right [resp. left] strong decomposer if
f (f ∗ (x)y) = f (y) [resp. f (xf∗ (y)) = f (x)]
: ∀x, y ∈ G.
(b) right [resp. left] semi-strong decomposer if
f (f ∗ (x)y) = f (f ∗ (e)y) [resp. f (xf∗ (y)) = f (xf∗ (e))]
: ∀x, y ∈ G.
(c) right [resp. left] decomposer if
f (f ∗ (x)f (y)) = f (y) [resp. f (f (x)f∗ (y)) = f (x)]
: ∀x, y ∈ G.
(d) right [resp. left] weak decomposer if
f (f ∗ (e)f (x)) = f (x) , f (f ∗ (x)f (e)) = f (e)
: ∀x ∈ G.
[resp. f (f (x)f∗ (e)) = f (x) , f (f (e)f∗ (x)) = f (e)
∗
: ∀x ∈ G].
(e) right [resp. left] separator if f (G) ∩ f (G) = {f (e)} [resp. f (G) ∩ f∗ (G)
= {f (e)}].
A STUDY OF DECOMPOSER AND RELATED ...
99
In each of the above parts if f (e) = e, then we will add the word standard to the
titles. For example “f is a standard right separator” means f ∗ (G)∩f (G) = {e}.
We call f a decomposer or a two-sided decomposer [resp. a separator] if it is a
left and right decomposer [resp. separator].
Example 1.1. Consider G = {1, a, a2 , a3 , b, ba, ba2 , ba3 } ∼
= D4 (a4 = b2 =
−1
3
2
3
1, bab = a = a ). Put Ω = {1, ba, ba , ba } and
(
x x∈Ω
f (x) =
bx x ∈
/ Ω.
Considering the relation x ∈
/ Ω ⇔ bx ∈ Ω, it can be seen that f is a (standard)
right strong decomposer.
Now, consider the additive group R and fix b ∈ R \ {0}. For a real number a
denote by [a] the largest integer not exceeding a and put (a) = a − [a] (the
decimal part of a). Now, set
a
a
[a]b = b[ ] , (a)b = b( ).
b
b
We call [a]b the b-integer part of a and (a)b the b-decimal part of a. Also [ ]b ,
( )b are called b-decimal part function and b-integer part function, respectively.
The b-parts functions are decomposers. Moreover, the b-decimal part function
( )b is a strong decomposer (see [3,4]). Also, for every constant real number c,
the function f := ( )b + c is a semi-strong decomposer.
It is shown in [2,4] that
(a) f is a right strong decomposer ⇒ f is a right semi-strong decomposer ⇒ f
is a right decomposer ⇒ f is a right weak decomposer.
Note that the converses of the above implications are not necessarily true. For
example, the real function f := ( )1 + 12 is a (right) semi-strong decomposer
but it is not a (right) strong decomposer.
(b) f is a standard right strong decomposer ⇔ f is a standard right semi-strong
decomposer ⇒ f is a standard right decomposer ⇒ f is a standard right weak
decomposer ⇒ f is a standard right separator.
(c) If f is a right strong decomposer, then f is a right separator, an idempotent,
f f ∗ = f (e) and
f ∗ (e).f f ∗ = f ∗ f = e
,
hf (e)i 6 f ∗ (G) 6 G.
Similar properties hold for the left case. Also, a left or right decomposer function is a (two-sided) strong decomposer if and only if f ∗ (G) = f∗ (G) E G.
100
2.
M. H. HOOSHMAND
Relations to Local Homomorphisms
Below introduces a type of local (partial) homomorphisms whic is very useful
for the topic.
Definition 2.1. Assume ∆, Ω are nonempty subsets of G. We call a map f from
G to G a “∆ × Ω-homomorphism” if f (δω) = f (δ)f (ω), for all δ ∈ ∆, ω ∈ Ω.
If f is a ∆ × Ω-homomorphism and e ∈ ∆ ∪ Ω, then f (e) = e. A function f is
an endomorphism if and only if it is a G × G-homomorphism.
Now, let g,h be arbitrary functions from G to G. The function f is a g(G) ×
h(G)-homomorphism if and only if
f (g(x)h(y)) = f g(x)f h(y) ;
∀x, y ∈ G.
Hence f is a f ∗ (G) × f (G)-homomorphism if and only if
f (f ∗ (x)f (y)) = f f ∗ (x)f 2 (y) ;
∀x, y ∈ G,
and it is a f ∗ (G) × G-homomorphism if and only if
f (f ∗ (x)y) = f f ∗ (x)f (y) ;
∀x, y ∈ G.
Also, if f is an idempotent f (G) × G-homomorphism then
f (f (x)y) = f (x)f (y) ;
∀x, y ∈ G.
It is interesting to know that if f (e) = e, then the converse is also valid (notice
that without the condition f (e) = e, the converse is not true, e.g. if k ∈ Z, then
f (x) = [x]+k satisfies in the additive real numbers group, but f 2 6= f whenever
k 6= 0). It is easy to see that if f satisfies, then the following conditions are
equivalent:
(i) f (e) = e , (ii) f is an idempotent , (iii) f is a f (G) × G-homomorphism.
For if f satisfies and f (e) = e then f 2 (x) = f (f (x)e) = f (x)f (e) = f (x), for
all x ∈ G. So, f (f (x)y) = f (x)f (y) = f 2 (x)f (y), for all x ∈ G, which means
(iii) holds. Also, if f is a f (G) × G-homomorphism then putting x = y = e we
obtain f 2 (e) = f 2 (e)f (e) and so f (e) = e.
Finally, f satisfies if and only if f∗ is a right strong decomposer (see [2; p.
549]). Similar properties hold for:
f (xf (y)) = f (x)f (y) ;
∀x, y ∈ G.
If f is is an idempotent f (G) × f (G)-homomorphism then
f (f (x)f (y)) = f (x)f (y) ;
∀x, y ∈ G.
A STUDY OF DECOMPOSER AND RELATED ...
101
If f 2 = f , then the converse is also valid (e.g. if f (e) = e). For the above
functional equation the conditions f (e) = e and f 2 = f are not equivalent, for
if c > 0 is a fixed real number and f (x) = max{x, c}, then f satisfies in the
additive real numbers group and f 2 = f but f (0) = c 6= 0. It is clear that if
f satisfies, then f (G) is a sub-semigroup of G but (in general) f (G) is not a
subgroup of G (e.g. f (x) = max{x, c}).
Example 2.2. The functions ( )b and [ ]b are bZ × R-homomorphisms. But
they are not endomorphisms.
Notation. By H 6̇G we mean H is a sub-semigroup of G. Also, we put fe :=
f − (e) · f .
It can be shown that if f is a type of right decomposer (weak, ordinary, semistrong or strong), then fe is a standard form of the same type of decomposer.
Lemma 2.3. Let f : G → G.
(a) If f is a right decomposer and f ∗ (G)6̇G, then it is a right strong decomposer.
(b) If f is a right weak decomposer and fe is a fe∗ (G) × fe (G)-homomorphism,
then f is a right decomposer.
(c) If f is a standard right separator and f f ∗ = f ∗ f , then it is a standard right
weak decomposer.
Proof. (a) Since f ∗ (x)f ∗ (y) ∈ f ∗ (G), then
f (f ∗ (x)y) = f (f ∗ (x)f ∗ (y)f (y)) = f (y) :
∀x, y ∈ G.
(b) fe is a standard right weak decomposer and so fe fe∗ = e, fe2 = fe . Hence,
we have
fe (fe∗ (x)fe (y)) = fe fe∗ (x)fe2 (y) = fe (y) :
∀x, y ∈ G.
Therefore fe is a standard right decomposer, so f is a right decomposer.
(c) Fix x ∈ G. If c = f (f ∗ (x)) = f ∗ (f (x)), then c ∈ f ∗ (G) ∩ f (G) so c = e.
Therefore f f ∗ = f ∗ f = e so f is a standard right weak decomposer. Recall from [2] that if ∆ and Ω are subsets of G, then the notation A = ∆ · Ω
means A = ∆Ω and if δ1 ω1 = δ2 ω2 where δ1 , δ2 ∈ ∆, ω1 , ω2 ∈ Ω, then δ1 = δ2
and ω1 = ω2 . If A = ∆ · Ω, then we say A is direct product of (subsets) ∆ and
Ω. By the notation A = ∆ : Ω we mean A = ∆ · Ω and ∆ ∩ Ω = {e} and say A
is standard direct product of ∆ and Ω. If A = ∆.Ω, then ∆ [resp. Ω] is called
left [resp. right] factor of A. Note that additive notations are ∆ u Ω (direct
sum of subsets) and ∆+̈Ω (standard direct sum of subsets).
Clearly if ∆Ω = ∆ · Ω, then |∆Ω| = |∆||Ω| = |Ω∆|. Also if ∆ and Ω are
102
M. H. HOOSHMAND
nonempty subsets of G, then ∆Ω = ∆ · Ω if and only if (∆−1 ∆) ∩ (ΩΩ−1 ) = {e}
(in additive notation (∆ − ∆) ∩ (Ω − Ω) = {0}). Moreover, if ∆ and Ω are
finite, then ∆Ω = ∆ · Ω if and only if |∆Ω| = |∆||Ω|. If ∆Ω = ∆ · Ω and ∆ ∩ Ω
has an element that commutes with every element of ∆ ∩ Ω, then |∆ ∩ Ω| = 1.
Especially if ∆Ω = ∆ · Ω and e ∈ ∆ ∩ Ω, then ∆Ω = ∆ : Ω. If G = ∆.Ω, then
G = ∆e : Ωe where ∆e = ∆δ0−1 , Ωe = ω0−1 Ω and e = δ0 ω0 . One may see more
information about factorization of a group by its subsets in [8].
Example 2.4. Consider the additive real numbers group R and put Rb =
b[0, 1) = {bd|0 6 d < 1} , hbi = bZ where b 6= 0 is a constant real number.
We have R = hbi+̈Rb (see [2]). But the natural numbers set is not a factor
(subset) of R. Also, S3 = hσi : hτ i where σ and τ are the elements of order two
and three, respectively, but S3 is not decomposable by its non-trivial (normal)
subgroups.
Projections. Let G = ∆ · Ω. Define the functions P∆ , PΩ , from G to G,
by P∆ (x) = δ, PΩ (x) = ω, where x = δω, δ ∈ ∆, ω ∈ Ω. Clearly, they are
well-defined and P∆ (G) = ∆, PΩ (G) = Ω, PΩ∗ = P∆ . We call PΩ , [resp. P∆ ]
right [resp. left] projection.
Example 2.5. The b-parts functions are projections of the direct decomposition R = hbi+̈Rb .
Now, we can state many equivalent conditions for a function f : G → G to be
a right weak, an ordinary, a semi-strong and a strong decomposer.
Theorem 2.6. Every conditions in each part (a),...,(f ) are equivalent.
(a)
i) f is a right decomposer.
ii) G = f ∗ (G).f (G).
iii) fe is a standard right decomposer.
iv) fe is a standard right weak decomposer
and a fe∗ (G) × fe (G)-homomorphism.
v)f ∗ (f ∗ (x)f (y)) = f ∗ (x), ∀x, y ∈ G.
(b)
i)f is a standard right decomposer.
ii) G = f ∗ (G) : f (G).
iii)f is a standard right weak decomposer
and f ∗ (G) × f (G)-homomorphism.
iv) f is a standard right weak decomposer
and f ∗ is a f ∗ (G) × f (G)-homomrphism.
A STUDY OF DECOMPOSER AND RELATED ...
(c)
i) f is a right strong decomposer.
ii) f is a right decomposer and f ∗ (G)6̇G.
iii)f is a right decomposer and f ∗ (G) 6 G.
iv) fe∗ is a fe∗ (G) × G-homomorphism and e ∈ f ∗ (G).
v) fe is a fe∗ (G) × G-homomorphism
and fe∗ is an idempotent, f (e) ∈ fe∗ (G).
vi) fe∗ is a fe∗ (G) × G-homomorphism and
an idempotent, f (e) ∈ fe∗ (G).
vii)f ∗ (f ∗ (x)y) = f ∗ (x)f ∗ (y), ∀x, y ∈ G.
(d)
i) f is a standard right strong decomposer.
ii) f is a right strong decomposer and
f ∗ is an idempotent.
iii)f is a right strong decomposer and
a f ∗ (G) × G-homomorphism.
iv)f is a f ∗ (G) × G-homomorphism
and f ∗ is an idempotent.
v)f ∗ is a f ∗ (G) × G-homomorphism
and an idempotent.
(e)
i) f is a standard right weak decomposer.
ii)f and f ∗ are idempotent.
iii) f is an idempotent and f f ∗ = f ∗ f .
iv) f ∗ is an idempotent and f f ∗ = f ∗ f .
v)f is a right separator and f f ∗ = f ∗ f .
vi)f f ∗ = f ∗ f = e.
vii)f f ∗ = f ∗ f = c, for some c ∈ G.
(f )
i)f is a right semi-strong decomposer.
ii)f (f ∗ (e)f (x)y) = f (xy), ∀x, y ∈ G
(i.e. f is a left semi-canceler ).
iii)f ∗ (e) · f is a standard right strong
decomposer.
103
104
M. H. HOOSHMAND
iv)f is a right decomposer and f ∗ (G)f (e)6̇G.
v)f is a right decomposer and f ∗ (G)f (e) 6 G.
vi) There exists a standard right strong
decomposer g and c ∈ G such that f = c · g.
vii)f ∗ (f ∗ (x)y) = f ∗ (x)f (e)f ∗ (f ∗ (e)y),
∀x, y ∈ G.
Moreover, if f is an endomorphism, then f is a standard right strong decomposer ⇔ f is a right decomposer ⇔ f is a right weak decomposer ⇔ f is a
right separator ⇔ f is an idempotent ⇔ f ∗ or f∗ is an idempotent.
Proof. (a) Let f be a right decomposer. If δ1 ω1 = δ2 ω2 , where δi ∈ f ∗ (G) and
ωi ∈ f (G), then
ω1 = f (δ1 ω1 ) = f (δ2 ω2 ) = ω2 ,
so δ1 = δ2 . Therefore G = f ∗ (G).f (G). Now let G = f ∗ (G).f (G). The relation
f ∗ (x)f (y) = f ∗ (f ∗ (x)f (y))f (f ∗ (x)f (y)),
implies f is right a decomposer. Also, the above relation shows that (i),(v) are
equivalent. The parts (i), (iii) are equivalent, obviously. Now, if f is a right
decomposer, then fe is a standard right decomposer and so fe fe∗ = e, fe2 = fe .
Thus fe is fe∗ (G) × fe (G)-homomorphism. Therefore (i) implies (iv). Now if
(iv) holds, then fe a is right decomposer, so f is a right decomposer.
(b) The part (a) implies (b), clearly.
(c) Lemma 2.3 implies that (i),(ii) and (iii) are equivalent and (i) implies (iv).
Now, let (iv) holds and put c = f (e)−1 (hence fe = c.f ). Since e ∈ f ∗ (G), then
f (e) ∈ fe∗ (G) = f ∗ (G)f (e) so c ∈ fe∗ (G). So
f (f ∗ (x)y) = c−1 cf (f ∗ (x)c−1 cy) = c−1 fe (fe∗ (x)cy) = c−1 fe (y) = f (y),
for all x, y ∈ G. Therefore (i), (iv) are equivalent. Also, the parts (i),(vii) are
equivalent, clearly.
Since f (e) ∈ fe∗ (G) if and only if e ∈ f ∗ (G), then the other parts are equivalent,
similarly.
(d) If f ∗ is idempotent, then f is a standard right strong decomposer if and
only if f is a f ∗ (G) × G-homomorphism. Considering this fact (c) implies (d).
(e) Considering the relations
2
f 2 = (f ∗ f )− .f , f ∗ = f ∗ .(f f ∗ )− ,
the parts (i),...,(vi) are equivalent. Now, let (viii) holds. Putting α = f (e),
β = f ∗ (e) we have c = f (β) = f ∗ (α). On the other hand f ∗ f = c implies
AASTUDY
STUDYOF
OFDECOMPOSER
DECOMPOSERAND
AND
RELATED
RELATED
... ...
105105
−1
ff22 =
= cc−1
.f.f so
soff22(β)
(β)==c−1
c−1
f f(β).
(β).Hence
Hence
2
∗
ff(c)
(c)==f f2 (β)
(β)==c−1
c−1
f (β)
f (β)
==
f ff∗f(α)
(α)
==
c =c =
f (β).
f (β).
Therefore
Thereforecc==e.e.
rightsemi-strong
semi-strongdecomposer,
decomposer,then
then
(f)
(f) IfIf ff isis aaright
∗
∗ ∗
ff(f(f∗ (e)f
(e)f(x)y)
(x)y)==f (f
f (f
(x)f
(x)f
(x)y)
(x)y)
==
f (xy),
f (xy),
f satisfies
(ii),(ii),
so
so itit isis aaleft
leftsemi-canceler
semi-canceler(see
(seethe
thenext
nextsection).
section).Conversely,
Conversely,
if fif satisfies
then
then
∗
−−
− −
∗
ff(f
(f∗∗(e)y)
(e)y)==ff(f(f∗ (e)f
(e)f
(x)f
(x)f
(x)y)
(x)y)
==
f (xf
f (xf
(x)y)
(x)y)
= f=(ff∗(f
(x)y),
(x)y),
rightsemi-strong
semi-strongdecomposer.
decomposer.Also
Also
wewe
have
have
thus
thus ff isis aaright
∗
∗ ∗ ∗
−1−1∗ ∗
∗
−1 −1
ff(f
(f∗∗(x)y)
(x)y)==ff(f(f∗ (e)y)
(e)y)⇔⇔f ∗f(f
(f(x)y)
(x)y)
f f(x)y
(x)y
==
f ∗f(f∗∗(f
(e)y)
(e)y)
f ∗ (e)y
f ∗ (e)y
∗ ∗∗
∗
∗ ∗∗ ∗
⇔
⇔f f∗ (f
(f(x)y)
(x)y)==f ∗f(x)f
(x)f
(e)f
(e)f
(f (f(e)y).
(e)y).
∗
Considering
Consideringthe
theabove
aboverelations
relationsand
andf ∗f =
=
fe∗f.fe∗ .f
(e)(e)
and
and
Lemma
Lemma
2.3,2.3,
we we
cancan
con-conclude
clude other
otherparts
partsofof(f).
(f).
∗
∗
endomorphism,then
thenf fis is
f ∗f(G)×G
(G)×G
and
and
f ∗f(G)×f
(G)×f
(G)-homomorphism
(G)-homomorphism
Finally,
Finally,ififffisisendomorphism,
∗∗
∗∗
Thereforethe
thelast
lastpart
part
ofof
the
the
theorem
theorem
is concluded
is concluded
from
from
(e),(a)
(e),(a)
and
andffff ==ff ff. .Therefore
and
and (d).
(d). 
andg gbebetwo
tworight
rightdecomposer
decomposer
functions
functions
with
with
the the
Corollary
Corollary 2.7.
2.7. (i)
(i)Let
Letf fand
∗
∗
same
same ∗-range
∗-range (i.e.
(i.e. ff∗ (G)
(G)==g ∗g(G)).
(G)).Then
Thenf fis isa right
a right
strong
strong
decomposer
decomposer
if if
rightstrong
strongdecomposer.
decomposer.Moreover,
Moreover,
if fif or
f or
g isg aisright
a right
strong
strong
and
and only
only ififggisisaaright
= g=and
g and
decomposer,
decomposer, then
then(they
(theyare
areright
rightstrong
strongdecomposer
decomposer
and)
and)
f gf=g f=, fgf, gf
|f
|f(G)|
(G)|==|g(G)|.
|g(G)|.
G ==∆∆· ·ΩΩ11==∆∆· Ω
·Ω
and∆∆
̇G
̇Goror0 =
0 =
|∆||∆|
<<
∞,∞,
then
then
|Ω1|Ω
|=
= 2|Ω
| (for
(ii)
(ii) IfIf G
2 2and
1 | |Ω
2 | (for
andwewehave
have
[0,[0,
1) 1)
∼∼
[1,[1,
2) 2)
). ).
example
example RR==ZZ+̈[0,
+̈[0,1)1)==ZZ
+̇[1,
+̇[1,2)2)and
f and
g g
Proof.
Proof. The
The first
firstpart
partofof(i)(i)isisconcluded
concludedfrom
from
Theorem
Theorem
2.6.2.6.
Now,
Now,
if fif and
∗
∗
(G)
==
g ∗g(G),
(G),
then
then
forfor
every
every
x ∈xG∈ G
are
are right
right strong
strongdecomposers
decomposersand
andf ∗f(G)
∗
∗
∗ ∗
xx==f f∗ (x)f
(x)f
(x)
(x)==g ∗g(x)f
(x)f
(g(x))f
(g(x))f
(g(x)).
(g(x)).
∗
(x)f∗∗(g(x))
(g(x))∈∈ff∗ (G),
(G),then
thenf (x)
f (x)==
f g(x)
f g(x)
(by(by
Theorem
Theorem
2.6).
2.6).
Therefore,
Therefore,
Since
Since gg∗∗(x)f
ff =
= ffgg and
andsimilarly
similarlygg==gfgf. .Putting
PuttingΩfΩ=
=
f
(G)
f
(G)
and
and
Ω
Ω
=
g(G),
=
g(G),
we
we
have
have
f
g g
ΩΩff==ff(g(G))
(g(G))==f (Ω
f (Ω
g(fg(f
(G))
(G))
==
g(Ω
g(Ω
g )g ) , , ΩgΩ=
g =
f ).f ).
|Ωff| ||Ω
|Ωg |g |and
and|Ω|Ω
| |Ω|Ω
| so|Ω|Ω
|=
|Ω|Ω
Therefore
Therefore|Ω
g |g
f |fso
f | f=
g |.g |.
|∆||Ω
|=
|∆||Ω
|∆||Ω
| implies
|Ω1|Ω
|=
= 2|Ω
|. 2 |.
(ii)
(ii) IfIf ∆
∆ isis non-empty
non-emptyand
andfinite,
finite,then
then|∆||Ω
1 | 1=
2 | 2implies
1 | |Ω
106
M. H. HOOSHMAND
Also if ∆6̇G, then putting f = PΩ1 and g = PΩ2 (where PΩ1 and PΩ2 are
projections, see next page) we have f ∗ (G) = g ∗ (G) = ∆ 6 G, by Theorem 2.6,
so the part (i) implies |Ω1 | = |f (G)| = |g(G)| = |Ω2 |. Corollary 2.8. f is a left strong decomposer ⇔ f∗ is a right decomposerand
f∗ (G) 6 G.
Remark 2.9. The most important property of right decomposer functions is
G = f ∗ (G).f (G). In fact this property connects these functions to “decomposition of a group by its subsets.” For example the decomposition Z = hni+̈Zn ,
by a subgroup and a subset, is produced by the strong decomposer function ( )n ,
for every nonzero integer n (notice that Z is not decomposable with its two
non-trivial subgroups).
2.1
Existence of decomposer types functions
Up to now we have studied the properties of decomposer type functions. Now,
we show that how we can construct them and prove their existence, in arbitrary
groups.
Definition 2.1.1. Let ∆ ⊆ G. We say G is left [resp. right] ∆-decomposable if
∆ is a left [resp. right] factor of G, which means G = ∆ · Ω [resp. G = Ω · ∆],
for some Ω ⊆ G.
Let ∆ ⊆ G. If ∆ is singletons, then G is left and right ∆-decomposable. Also
G is left and right G-decomposable (trivially). Now consider the additive group
G = R and fix b ∈ R \ {0}. Then R is Rb -decomposable and R is not Ndecomposable, i,e, the real numbers group does not have natural part property
(although it has the integer part property, see Example 2.4).
Remark 2.1.2. Considering Theorem 2.6(c), if ∆6̇G but ∆ G, then G is not
left and right ∆-decomposable. Therefore, R is not (M, ∞)-decomposable for
all M > 0. Now, if ∆ 6 G, then G is (standard) left and right ∆-decomposable.
Moreover, if ∆ E G, then G is (standard two-sided) ∆-decomposable (see Remark 2.7 of [2]).
Now, we are ready to prove the existence of a vast class of decomposer type
functions.
Lemma 2.1.3. If G 6= 0 is a group (finite or infinite) for which |G| is not
a prime number, then nontrivial (standard) right and left strong decomposer
functions exist (and vice versa). In addition if G is not a simple group, then
nontrivial (standard) strong decomposer functions exist.
Proof. The hypothesis implies there exists non-trivial subgroup ∆ of G. So
A STUDY OF DECOMPOSER AND RELATED ...
107
Remark 2.9 gives us a subset Ω of G such that G = ∆ : Ω. Putting f = PΩ ,
we have f ∗ (G) = P∆ (G) = ∆ 6 G, thus Theorem 2.6 implies f is a nontrivial
left standard strong decomposer function on G. Moreover, if G is not simple
group, then there exists such a nontrivial subgroup ∆ which is normal. In this
case we claim that the function f defined by f = PΩ is a (two-sided) strong
decomposer. Considering the first part it is enough to show that it is a left strong
decomposer. For every y there exist y 0 , y 00 such that y −1 f ∗ (y) = f ∗ (y 0 )y −1 and
so
f (xf∗ (y)) = f (xy −1 f ∗ (y)y) = f (xf ∗ (y 0 )y −1 y)
00
= f (xf ∗ (y 0 )) = f (f ∗ (y )x) = f (x).
Therefore the proof is complete.
Note that there are so many left and right strong decomposer [strong associative] functions in group [non-simple group] G such that 1 6= |G| =
6 p, for every
prime numbers p. For example the cardinal number of all strong decomposer
real functions is 2c = |2R |, i.e. there exists a one to one correspondence between
the solutions set of b-decimal part functional equation f (x + y − f (y)) = f (x)
and all real functions!.
Acknowledgements
This work is supported by the Research Foundation of Islamic Azad UniversityShiraz Branch [grant number P 93/156].
References
[1] N. Brillouët-Belluot, Multiplicative symmetry and related functional equations, Aequationes Math., 51 (1996), 21-47.
[2] M. H. Hooshmand and H. Kamarul Haili, Decomposer and Associative
Functional Equations, Indag. Mathem., N. S., 18 (2007), 539-554.
[3] M. H. Hooshmand, Parter, periodic and coperiodic functions on groups
and their characterization, J. Math. Extension, 7 (2013), 1-14.
[4] M. H. Hooshmand, Decomposer Type Functions on Groups, British J.
Math. Comp. Sci., 4 (2014), 1835-1842.
[5] M. Kuczma, An Introduction to the Theory of Functional Equations
and Inequalities: Cauchy’s Equation and Jensen’s Inequality, Uniwersitet
Slaski, Poland (1985).
[6] Y. Matras, Sur l’équation fonctionnelle: f (x · f (y)) = f (x) · f (y), Acad.
Roy. Belg. Bull. Cl. Sci., 5 (1969), 731-751.
98
108
M. H. HOOSHMAND
S. ALIKHANI
AND N. GHANBARI
type
of M.
local
homomorphisms
we call
∆ × Ω-homomorphisms.
[7] T.
Rassias,
Functionalwhich
Equations
andthem
Inequalities,
Springer (2000).Let
us recall decomposer type functions on groups (from [2,4]) first, and then in[8] S. Szabo
andconceptions.
A. D. Sands,
Groupsweinto
Subsets,
Presstroduce
the new
ByFactoring
applying them,
obtain
manyCRC
new results
Taylor
and
Francis
Group,
Boca
Raton,
2009.
regarding to decomposer type functions on groups and prove some related theorems.
Let (G, .) be a group with the identity element e. If f and g are functions from
G to G, then define
functions f.g and f − by
Mohammad
Hadi the
Hooshmand
Department of Mathematics
= Mathematics
f (x)g(x) , f − (x) = f (x)−1 :
∀x ∈ G.
f.g(x) of
Assistant Professor
Shiraz Branch, Islamic Azad University
We denote the identity function on G by ιG and put f ∗ = ιG .f − , f∗ = f − .ιG
Shiraz, Iran
and call f ∗ [resp. f∗ ] left ∗-conjugate of f [resp. right ∗-conjugate of f ]. They
E-mail: [email protected]
are also called ∗-conjugates of f . Note that f ∗ (e) = f∗ (e) = f − (e) = f (e)−1 .
If (G, +) is additive group, then the notations e, f − , f.g, f.g − are replaced by
0, −f , f + g, f − g and we have f ∗ = f∗ = ιG − f . It is easy to see that
f is idempotent ⇔ f ∗ f = e ⇔ f∗ f = e
2
f ∗ = f ∗ ⇔ f f ∗ = e ⇔ f ∗∗ f ∗ = e , f∗2 = f∗ ⇔ f f∗ = e ⇔ f∗∗ f∗ = e.
Also, if f is endomorphism then f − f = f f − , f ∗ f = f f ∗ and f∗ f = f f∗
(i.e. the compositions of f and its ∗-conjugations are commutative).
Now, we recall from [2,4] decomposer type functions as follows.
A function f from (a group) G to G is called:
(a) right [resp. left] strong decomposer if
f (f ∗ (x)y) = f (y) [resp. f (xf∗ (y)) = f (x)]
: ∀x, y ∈ G.
(b) right [resp. left] semi-strong decomposer if
f (f ∗ (x)y) = f (f ∗ (e)y) [resp. f (xf∗ (y)) = f (xf∗ (e))]
: ∀x, y ∈ G.
(c) right [resp. left] decomposer if
f (f ∗ (x)f (y)) = f (y) [resp. f (f (x)f∗ (y)) = f (x)]
: ∀x, y ∈ G.
(d) right [resp. left] weak decomposer if
f (f ∗ (e)f (x)) = f (x) , f (f ∗ (x)f (e)) = f (e)
: ∀x ∈ G.
[resp. f (f (x)f∗ (e)) = f (x) , f (f (e)f∗ (x)) = f (e)
: ∀x ∈ G].
(e) right [resp. left] separator if f (G) ∩ f (G) = {f (e)} [resp. f (G) ∩ f∗ (G)
= {f (e)}].
∗