Article Welcome To ITB Journal A11157

ITB J. Sci., Vol. 44 A, No. 3, 2012, 204-216

204

Boolean Algebra of C-Algebras
G.C. Rao1 & P. Sundarayya2
1

2

Department of Mathematics, Andhra University, Visakhapatnam-530 003, India
Department of Mathematics, GIT, GITAM University, Visakhapatnam-530 045, India
E-mails: gcraomaths@yahoo.co.in; psundarayya@yahoo.co.in
Abstract. A C- algebra is the algebraic form of the 3-valued conditional logic,
which was introduced by F. Guzman and C.C. Squier in 1990. In this paper,
some equivalent conditions for a C- algebra to become a boolean algebra in
terms of congruences are given. It is proved that the set of all central elements
B(A) is isomorphic to the Boolean algebra BS ( A) of all C-algebras Sa, where a
B(A). It is also proved that B(A) is isomorphic to the Boolean algebra BR ( A)
of all C-algebras Aa, where a


B(A).

Keywords: Boolean algebra; C-algebra; central element; permutable congruences.

1

Introduction

The concept of C-algebra was introduced by Guzman and Squier as the variety
generated by the 3-element algebra C={T,F,U}. They proved that the only
subdirectly irreducible C-algebras are either C or the 2-element Boolean algebra
B ={T,F} [1,2].
For any universal algebra A, the set of all congruences on A (denoted by Con A)
is a lattice with respect to set inclusion. We say that the congruences , are
permutable if  =  . We say that Con A is permutable if  = 
for all ,
Con A . It is known that Con A need not be permutable for any Calgebra A.
In this paper, we give sufficient conditions for congruences on a C-algebra A to
be permutable. Also we derive necessary and sufficient conditions for a Calgebra A to become a Boolean algebra in terms of the congruences on A. We
also prove that the three Boolean algebras B(A), the set of C-algebras BS ( A) and

the set of C-algebras BR ( A) are isomorphic to each other.

Received November 1st, 2011, Revised April 5th, 2012, Accepted for publication July 7th, 2012
Copyright © 2012 Published by LPPM ITB, ISSN: 1978-3043, DOI: 10.5614/itbj.sci.2012.44.3.1

Boolean Algebra of C-Algebras

2

205

Preliminaries

In this section we recall the definition of a C-algebra and some results from
[1,3,5] which will be required later.
Definition 2.1. By a C -algebra we mean an algebra < A, , , > of type
(2,2,1) satisfying the following identities [1].
(a)
(b)
(c)

(d)
(e)
(f)
(g)

x = x;
( x y)
x (y
x (y
( x y)
x (x
( x y)

= x y;
z ) = ( x y ) z;
z ) = ( x y) ( x z );
z = ( x z) ( x y z);
y ) = x;
( y x) = ( y x) ( x y).


Example 2.2. [1]:
The 3- element algebra C={T, F, U} is a C-algebra with the operations
and defined as in the following tables.

x

x

T
F
U

F
T
U

T
T
F
U


F

U

T F
F F
U U

U
F
U

T
T
F
U

F


T T T
T F U
U U U

Every Boolean algebra is a C-algebra.
Lemma 2.3. Every C-algebra satisfies the following laws [1,3,5].
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)

x x = x;
x y = x ( x y) = ( x y) x;
x ( x x) = x;
( x x ) y = ( x y) ( x y);

( x x ) x = x;

x x = x x;
x y x = x y;
x x y=x x;
x ( y x) = ( x y )

x.

The duals of all above statements are also true.

U

,

206

G.C. Rao & P. Sundarayya

Definition 2.4. If A has identity T for

( that is, x T = T x = x for all
x A ), then it is unique and in this case, we say that A is a C -algebra with T.
[1].
We denote T ' by F and this F is the identity for
Lemma 2.5 [1]: Let A be a C -algebra with T and x, y
(i)
(ii)
(iii)
(iv)

A. Then

x y = F if and only if x = y = F
if x y = T then x x = T .
x T= x x
x F=x x .

Theorem 2.6. Let < A, , , > be a C-algebra. Then the following are
equivalent [6]:
(i)

(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)

A is a Boolean algebra.
x ( y x) = x, for all x, y A .
x y = y x, for all x, y A .
( x y ) y = y , for all x, y A .
x x is the identity for , for every x A .
x x = y y , for all x, y A .
A has a right zero for .
for any x, y A , there exists a A such that x a = y
for any x, y A , if x y = y , then y x = x.

a = a.


Definition 2.7. Let A be a C-algebra with T. An element x A is called a
central element of A if x x = T . The set of all central elements of A is called
the Centre of A and is denoted by B(A) [5].
Theorem 2.8. Let A be a C-algebra with T .Then < B ( A), , , > is a Boolean
Algebra [5].

3

Some Properties of a C -algebra and Its Congruences

In this section we prove some important properties of a C-algebra and we give
sufficient conditions for two congruences on a C-algebra A to be permutable.
Also we derive necessary and sufficient conditions for a C-algebra A to become
a Boolean algebra in terms of the congruences on A.
Lemma 3.1. Every C-algebra satisfies the following identities:

Boolean Algebra of C-Algebras

(i) x

(ii) x

y = x (y
y = x (y

207

x );
x ).

Proof. Let A be a C-algebra and x, y

A. Now,

x y = x ( x y) = x ( x y x ) = [ x ( x x )] [ x y ( x ( x x ))] =
[ x ( x x )] [ x y x ( x x )] = [ x ( x x )] [ x y ( x x )] = ( x y )
( x x ) = x ( y x ). Similarly x y = x ( y x ).
Lemma 3.2. Let A be a C-algebra and x, y A. Then x y x = x y y .
Proof. Let A be a C-algebra and x, y A. By Lemma 2.3[b],[f] and Lemma
3.1, we have x y x = x (( y x ) y ) = [ x ( y x )] y = ( x y )
y= x y y= x y y.
Lemma 3.3. Let A be a C-algebra with T , x, y
x y = y x.

A and x

y = F . Then

Proof. Suppose that x y = F. Then F = x y = x ( x y) = ( x x )
( x y) = ( x x ) F = x x . now x y = F ( x y) = ( x y) ( x y)
= ( x x y) ( x y x y ) (By Def 2.1) = ( x y) ( x y x) (By
2.3[g]) = ( x x ) ( y x) = F ( y x) = y x.
In [6], it is proved that if A is a C-algebra with T then B( A) = {a A |
a a = T } is a Boolean algebra under the same operations , , in the Calgebra A. Now we prove the following.
Theorem 3.4. Let A be a C-algebra with T and a, b
a b B(A) . Then a B(A).

A such that

Proof. Let A be a C-algebra with T and a, b A such that a b B(A). Then
T = (a b) (a b) = (a b) (a b )
= (a b a ) (a b b ) = (a b a ) (a b a ) (By Theorem 3.2)
=a b a .
Therefore, T = a b a ...(I)

208

G.C. Rao & P. Sundarayya

Now, a a

Hence a

=
=
=
=

(a
(a
(a
a

a) T
a ) (a b a ) (by(I))
(a b a )) (a (a b a ))
(a (a b a ))
= a (a b a ) (By 2.3[b])
= a b a = T.

B( A).

The converse of the above theorem need not be true. For example, in the Calgebra C, F B(C ) but F U = U B(C). We have the following
consequence of the above theorem.

A and a b

Corollary 3.5. Let A be a C-algebra with T, a, b
a B( A).

Proof. Let a b B( A). Then we have, (a b)
a B( A) a B( A).
In [1], it is proved that if A is a C-algebra, then
a congruence on A and

x

=

x

x x

x

B( A)

= {( p, q) | x

B( A) . Then

a

b

p = x q} is

. In [6], if A is C-algebra with T then

is a factor congruence if and only if x B( A). They also proved that x ,
permutable congruences whenever both x, y
important properties of these congruences.

a b

Proof. (i) ( x, y)
a

=

; (ii)

a

x=a

b

b a



b

a b

.

y

b a b x=b a b x
b a x=b a x

( x, y)
Therefore

a b

b a

b a

. Similarly,

b a

a b.

are

A . Then we have the

a b

b

y

x

B( A) . Now we prove some

Theorem 3.6. Let A be a C-algebra with T and a, b
following (i)

B( A)

Hence

a b

=

b a

.

Boolean Algebra of C-Algebras

(ii) Let ( x, y)

( z, y )

a

a

 b . Then there exists z

. Thus b x = b z and a z = a

z=a b a z=a b a
a



b

a b

209

A such that ( x, z)

and

x=a b

y. Now, a b

y. Therefore, ( x, y )

y=a b

b

a b

. Thus

.

In the following we give an example of a C-algebra G without T in which the
Con A is not permute.
Example 3.7. Consider the C-algebra G = {a1, a2 , a3 , a4 , a5} where a1 = (T ,U ) ,
a2 = ( F , U ), a3 = (U , T ), a4 = (U , F ), a5 = (U ,U ) under pointwise operations
in C .

x

x

a1

a2

a2

a1

a2

a3

a4

a5

a1

a1

a2

a5

a5

a5

a1

a2

a2

a2

a2

a2

a2

a3

a4

a3

a5

a5

a3

a4

a5

a4

a3

a4

a4

a4

a4

a4

a4

a5

a5

a5

a5

a5

a5

a5

a5

a2

a3

a4

a5

a1
a2

a1
a1
a1

a1
a2

a1
a5

a1
a5

a1
a5

a3

a3

a3

a3

a3

a3

a4

a5

a5

a3

a4

a5

a5

a5

a5

a5

a5

a5

This algebra (G, , , ) is a C-algebra with out T.
Let = diagonal of A. Then we have the following:
a1

a3

= {( x, y ) | a1

x = a1

y}

=

{(a3 , a4 ), (a4 , a5 ), ( a5 , a3 ), ( a4 , a3 )( a5 , a4 ), ( a3 , a5 )}

=

{(a1 , a2 ), (a2 , a5 ), (a5 , a1 ), ( a2 , a1 )( a5 , a2 ), ( a1, a5 )}

210

G.C. Rao & P. Sundarayya

Now,

a1
a3

Therefore



a3



a1



a1

=

a1

=

a1

a3



a3

{(a4 , a1 ), (a4 , a2 )( a3 , a1 ), (a3 , a2 )}

a3

{(a2 , a4 ),(a2 , a3 )(a1 , a3 ),(a1 , a4 )}

a3

a1 .

Theorem 3.8. Let A a C-algebra with T and a

b A, a ,

b

permute and



a

b

=

a b

B( A) . Then for any

.

Proof. Let A be a C-algebra with T and a B(A) . By Theorem 3.6,
Now let ( p, q) a b . Then a b p = a b q b a b
a  b
a b.
p = b a b q b a p = b a q. Consider, r = (a p) (a q) .
Now a r = a [(a p) (a q)] = (a p ) (a a q ) = (a p ) ( F q )

= (a p) F a p. Therefore (r, p)
b [(a p) (a q)] = [b a p] [b a
=b

((a

=b

q . Therefore (q, r)

b



a

=

q)

a b

(a

q)) = b

. Thus

b



(( a

Now, b r =
q] = (b a q) (b a q)
q) = b (T q) (since a B( A))

a)

(r, q)

b

( p, r)

a

b

a.

. Thus ( p, q)

is a congruence on A and hence

a

permutable congruences and hence

a



b

=

b



a

=

a b

a



b

=

Proof. i) We know that if a
theorem

a



b

=

b



a

=

a b

; ii) a b B( A)

b
a b.

B(A) then a

a



a.
a

,

Hence
b

are

.

Corollary 3.9. Let A be a C-algebra with T and a, b

a b B( A)



b

b

=

A . Then i)

a b.

B(A) and hence by the above

Similarly, we can prove ii).

Let A be a C-algebra. If Con(A) is permutable, then A need not be a Boolean
algebra. For example, in the C-algebra C, the only congruences are , and
they are permutable. But C is not a Boolean algebra. Now we give equivalent
conditions for a C-algebra to become a Boolean algebra in terms of congruence
relations.
Theorem 3.10. Let ( A, , , ) be a C-algebra with T. Then the following are
equivalent. (i) Let ( A, , , ) be a Boolean algebra. (ii)

x

A. (iii)

x x

=

for all x

A.

x

x

=

for all

Boolean Algebra of C-Algebras

211

(2): Let A be a Boolean algebra and x

A . Let ( p, q) x
x.
Then x p x q and x p = x q. Now, p = ( x x ) p = ( x p)
. Therefore
( x q) = ( x q) ( x q) = ( x x ) q = q. Thus x
x
(iii). (iii) (i): Suppose
x
x = . Since
x
x = x x , we get (ii)
Proof. (1)

A. We prove that x  x = A A. Let ( p, q) A A .
Write t = ( x p) ( x q). Now, x t = x (( x p) ( x q)) = ( x p)
( x x q ) = ( x p) ( x x ) = x ( p x ) = x p. Also, x t = x
x x

(( x

=

for all x

p) ( x

q)) = x

q)) = ( x

x

p) ( x

q. Therefore ( p, t )

x

x

q)) = ( x

and (t , q )

x

x) ( x

q)) = ( x

. Thus ( p, q)

x

(x



x

.

Hence we get x  x = A A. Also x
x
x and
x = x x = . That is
are permutable factor congruences. Therefore, by Theorem 2.6, we have
x B( A). Thus A = B( A) and hence A is a Boolean algebra.

4

The C-algebra S x

We prove that, for each x

A , S x = {x t | t

A} is itself a C-algebra under

induced operations , and the unary operation is defines by ( x t )
x t'
. We observe that Sx need not be a subalgebra of A because the unary operation
in Sx is not the restriction of the unary operation on A. Also for each x A , the
set Ax = {x t | t A} is a C-algebra in which the unary operation is given by

( x t )* = x t . We prove that the B(A) is isomorphic to the Boolean algebra
BS ( A) of all C-algebras Sa where a

B(A) . Also, we prove that B(A) is

isomorphic to the Boolean algebra BR ( A) of all C-algebras Aa , a
Theorem 4.1. Let < A, , , > be a C-algebra, x

B( A) .

A and S x = {x t | t

Then < S x , , ,* > is a C-algebra with x as the identity for
are the operations in A restricted to Sx and for any x t

, where

A}.
and

S x , here ( x t )*

is x t .
Proof. Let t , r , s

A . Then ( x t ) ( x r ) = x (t

( x r) = x (t r) Sx . Thus
**

,

r ) S x and ( x t )

are closed in Sx. Also

Consider ( x t ) = x ( x t ) = x ( x

*

is closed in Sx.

t ) = x t . Now [( x t ) ( x r )]*

212

G.C. Rao & P. Sundarayya

= [ x (t

r )]* = x (t

consider

(x t) (x r)

x (t s) x (t

x r = ( x t )* ( x r )*.

r)=x t

( x s) x [(t r) s] = x

r s)

( x t ) ( x s)

Now,

(t s) (t

r s)

( x t ') ( x r) ( x s)

( x t ) ( x s ) ( x t ) ( x r ) ( x s ) . The remaining identities of a
C-algebra also hold in Sx because they hold in A. Hence, Sx is itself a C-algebra.
Also x is the identity for because x x t = x t = x t x . Here x x is
the identity for .
Theorem 4.2. Let A be a C-algebra. Then the following holds.
(i) S x = S y if and only if x = y ;
(ii) S x

Sy

Sx y ;

Sx = Sx x ;
(iv) ( S x ) x y = S x y .
(iii) S x

Proof. (i) Suppose S x = S y . Since x = x

x S x = S y and y = y

y

S y = S x . Therefore x = y t and y = x r for some t, r A . Now,
x = y t = ( y t y ) ( y y t ) = ( x y ) ( y x) = ( y x) ( x y ) =
( x r x) ( x x r ) = x r = y. The converse is trivial. (ii) Suppose
t S x S y . Then t = x s = y r for some s, r A. Now, t = x x s
=x t=x
we have Sx

y r

Sx

Sx

x

y x r|r

S x defined by

with kernel

x

Proof. Let t , r
and

x

x

S x = S x x . (iv) (S x ) x y = {x y t |
A} = {x y r | r A} = S x y
.

Theorem 4.3. Let A be a C-algebra with T and x

A

x =x

by (ii). Since x

Sx . Hence S x

Sx

x

t S x } = {x

S x y . (iii) S x

x

(t ) = x t for all t

and hence A/

A. Then

x

x

Sx.

(t

r) = x t

(t ) = x t = ( x t )* = (

x

A, then the mapping

x

:

A is a homomorphism of A to S x

r=x t

(t ))* . Clearly,

x

x r=

(t r ) =

x

x

(t )

(t )

x

x

(r )

(r ).

in S x. Therefore x
Also x (T ) = x T = x x , which is the identity for
is a homomorphism. Hence by the fundamental theorem of homomorphism
A/Ker x S x and Ker x = {(t , r ) A A | x (t ) = x (r )} = {(t , r ) A A |

Boolean Algebra of C-Algebras

x t = x r} = {(t , r )
hence A/

A A| x

t=x

r}

x

213

(by Lemma 2.3 [b]) = and

Sx.

x

Theorem 4.4. Let A be a C-algebra with T and a

B(A), then A

Sa S a .

Proof. Define
: A Sa Sa by ( x) = ( a ( x), a ( x)) for all x A.
Then, by Theorem 4.3,
is well-defined and
is a homomorphism. Now, we
( a ( x), a ( x))
prove that
is one-one. Let x, y A . Then ( x) = ( y)

=(

a

( y),

a

x=a

( y))

(a y ) =
x=a t,
a y=a
( x y) = (

=
=
=
=
Therefore,

(a x, a

x) = ( a

y, a

y)

a x=a

y

and
y. Now x = F x = (a a ) x = (a x) (a x) = (a y )
is onto. Let ( x, y) Sa Sa . Then
y. Finally, we prove that
and y = a r for some t, r A . Therefore, a x = x,
and
Now,
a y =T y =T
a x = T , a y = y.
a

( x y ), a ( x y ))
(a ( x y ), a ( x y ))
((a x) (a y ), (a x) (a
( x T , T y)
( x, y ).
a

is onto and hence

is an isomorphism. Therefore A

Lemma 4.5. Let A be a C-algebra. Then for a, b
(i) a b = b a if and only if Sa

y ))

b

= Sa

Sa S a .

A:

Sb

(ii) Sa b = Sup{Sa , Sb } in the poset ( {S x | x
The converse is not true.

A}, ), then a b = b a.

b = b a. Then clearly Sa b Sa Sb . By
Theorem 4.2(ii) Sa Sb Sa b . Hence Sa b = Sa Sb . Conversely assume
that Sa b = Sa Sb . Clearly a b Sa b = Sa Sb . Therefore a b Sb
a b = b t for some t A. Now b a = b a b = b b t = b t =
a b. (ii) Assume that a, b A and Sa b = Sup{Sa , Sb }. Then Sa b = Sb a
and hence a b Sa b = Sb a . Therefore a b = (b a) t for some t A.
Proof. (i) Suppose that a

214

G.C. Rao & P. Sundarayya

Now (b a) (a b) = (b a ) ((b a ) t ) = (b a ) t = a b. Similarly
we can prove that (a b) (b a ) = b a. Hence a b = b a. The
converse need not be true, for example for the C-algebra C ,
SU = {U }, ST = {T } and U T = T U . But SU T (= SU ) is not an upper
bound of {SU , ST }.
Now we prove BS ( A)

{Sa | a B( A)} is a Boolean algebra under set

inclusion.
Theorem

Let

4.6.

BS ( A) ={Sa | a

be

< A, , , >

a

C-algebra

with

T.

Then

B( A)} is a Boolean algebra under set inclusion.

Proof. Clearly (BS ( A) , ) is a partially ordered set under inclusion. First we
show for a, b

B( A) , S a

b

is the infimum of {Sa , Sb } and S a

supremum of {Sa , Sb } for all

B( A) . Let

a, b

B( A) . Then

a, b

a b = b a and a b = b a. Hence by the above Lemma 4.5, S a
infimum of {Sa , Sb } . Let t
t = a x = (a (a b)) x = (a
Sb

Sb

a

= S a b . Therefore S a

Sa . Then t = a x for some x

b

is the

A . Now
Similarly

(b a) x = (a b) a x Sa b .
b

is the

b

is an upper bound of S a , Sb . Suppose Sc is

an upper bound of S a , Sb . t

Sa b . Then t = (a b) x for some x A .
Now t = (a b) x = (a x) (a b x) = (a x) (b a x) Sc (since
a x Sa Sc , b a x Sb Sc and Sc is closed under ). Therefore
S a b is the supremum of {S a , Sb }. Denote the supremum of {Sa , Sb } by
S a S b and the infimum of {Sa , Sb } by S a S b . Now ST Sa = ST a = ST
and S F
greatest

Sa = S F

= S F . Therefore ST is the least element and S F is the
(BS ( A) , ) .
Now
for
any
element
of
a

a, b, c B( A),(Sa Sb ) Sc = S(a
Sc ) (Sb

Sc ).

Also,

Sa

b) c

= S( a

Sa = Sa

a

c ) (b c )

= ST

= S( a

c)

and

Sa

S( b

c)

= ( Sa

Sa = Sa

a

= SF .

Therefore (BS ( A) , ) is a complimented distributive lattice and hence it is a
Boolean algebra.

Boolean Algebra of C-Algebras

Theorem 4.7. Let A be a C-algebra with T Define

(a) = sa for all a B(A). Then
Proof. Let a, b

(a b) = S(a
Clearly

b)

= Sa

: B ( A)

BS ( A ) by

is an isomorphism.

( a b) = S ( a

B( A) . Then

215

Sb = (a)

(b) ,

is both one-one and onto. Hence B( A)

b)

= Sa

Sb = ( a )

(b).

(a ) = Sa = ( Sa ) = ( (a)) .
BS ( A ) .

In [3] we defined a partial ordering on a C-algebra by x y if and only if
y x = x and we studied the properties of this partial ordering. We gave a
number of equivalent conditions in terms of this partial ordering for a C-algebra
to become a Boolean algebra. In [4] we proved that, for each x A ,
Ax = {s A | s x} is itself a C-algebra under induced operations , and
the unary operation is defined by s* = x

s we also observed that Ax need not

be an algebra of A because the unary operation in Ax is not the restriction of

A, we proved that Ax is isomorphic to the

the unary operation. For each x

p = x q} . We can
easily see that the C-algebras S x , Ax are different in general where x A.
quotient algebra A/

x

where

x

= {( p, q) A A | x

Now, we prove that the set of all Aa 's where a B(A) is a Boolean algebra
under set inclusion. The following theorem can be proved analogous to
Theorem 4.6.
Theorem 4.8. Let A be a C-algebra with T. Then BR( A) :={Aa | a
Boolean Algebra under set inclusion in which
{Aa , Ab } = Aa b and the infimum of {Aa , Ab} = Aa b .

the

B( A)} is a

supremum of

The proof of the following theorem is analogous to that of Theorem 4.7.
Theorem 4.9. Let A be a C-algebra with T Define f : B( A)

BR ( A) by

f (a ) = Aa for all a B(A). Then f is an isomorphism.
The following corollary can be proved directly from Theorems 4.7 and 4.9.

216

G.C. Rao & P. Sundarayya

Corollary 4.10. Let A be a C-algebra with T. Then BR( A) , B( A) and BS ( A)
are isomorphic to each other.

References
[1]
[2]
[3]
[4]
[5]
[6]

Guzman, F. & Squier, C., The Algebra of Conditional Logic, Algebra
Universalis, 27, pp. 88-110, 1990.
Swamy, U.M., Rao, G.C., Sundarayya, P. & Kalesha Vali, S., Semilattice
Structures on a C-algebra, Southeast Asian Bulletin of Mathematics, 33,
pp. 551-561, 2009.
Rao, G.C., Sundarayya, P., C-algebra as a Poset, International Journal of
Mathematical Sciences, Serials Pub., New Delhi, 4(2), pp. 225-236, Dec.
2005.
Rao, G.C., Sundarayya, P., Decompositions of a C-algebra, International
Journal of Mathematics and Mathematical Sciences, Hindawi Pub. Cor.
U.S.A., 2006(3), Article ID 78981, pp. 1-8, 2006.
Swamy, U.M., Rao,G.C. & Ravi Kumar, R.V.G., Centre of a C-algebra,
Southeast Asian Bulletin of Mathematics, 27, pp. 357-368, 2003.
Stanely, B. & Sankappanavar, H.P., A Course in Universal Algebra,
Springer Verlag, 1981.

Dokumen yang terkait

MANAJEMEN STRATEGI RADIO LOKAL SEBAGAI MEDIA HIBURAN (Studi Komparatif pada Acara Musik Puterin Doong (PD) di Romansa FM dan Six To Nine di Gress FM di Ponorogo)

0 61 21

The Efficacy Of Glass Ionomer Hybrid As Orthodontic Brackets Adhesives To Inhibit Calsium Released Of The Ename

1 51 6

Evaluasi Pengendalian Internal Dalam Menunjang Efektivitas Sistem Pemberian Kredit Usaha Mikro Kecil Menengah (Studi Kasus Bank UMKM Cabang Jember); Evaluation Of Internal Control To Effectiveness Credit System Umkm (A Case Study of Bank UMKM Jember Branc

0 17 8

FAKTOR SOSIAL BUDAYA DAN ORIENTASI MASYARAKAT DALAM BEROBAT (Studi Atas Perilaku Berobat Masyarakat Terhadap Praktek Dukun Ponari) SOCIO-CULTURAL FACTORS AND SOCIETAL ORIENTATION IN THE TREATMENT (Study Of The Treatment Behavior Of Society To Ponari Medic

1 30 13

An Error Analysis On Students’ Ability In Using Gerund And To Infinitive

2 28 70

The Thematic Roles In Article Of 'What Ever Happened In Kraras, Timor Leste 'Park' Prabowo?' On The Jakarta Post Online December 20, 2013

0 15 0

Pengaruh Proce To Book Value,Likuiditas Saham dan Inflasi Terhadap Return Saham syariah Pada Jakarta Islamic Index Periode 2010-2014

7 68 100

The Ways In Which Information Given Related To The Type Of Visitor (A Description on Guiding Technique at the Room of History of Life in Bandung Museum of Geology)

0 16 48

Pengembangan Sistem Informasi Pengelolaan Tugas Akhir (Sipintar) Di Fakultas Teknik Mesin Dan Dirgantara ITB

0 53 234

Pelaksanaan deposito berjangka dalam upaya meningkatkan sumber dana bank pada Bank BNI Cabang ITB Bandung

1 12 55