Scientiae Mathematicae
Vol.1, No. 3(1998), 373{381
373
IDEAL THEORY OF BCC{ALGEBRAS
Jiang Hao
Received October 9, 1997
Abstract.
We introduce the concept of ideal in a BCC-algebra and prove some related prop-
erties. We also give a method for constructing a proper BCC-algebra by the extension of a
BCK-algebra with a small atom.
In 1984 Y. Komori introduced in [1] a notion of BCC-algebra. The relationship between
BCC-algebras and BCK-algebras are discussed in [3] and [4]. For basic properties of BCKalgebras the reader is referred to [2] and [3].
In this paper we discuss the further relationship between BCC-algebras and BCKalgebras and introduce the concept of ideal in a BCC-algebra. Some related properties
are proved.
x1. Preliminaries.
; 0) of type (2, 0) is called a BCC-algebra if it satises
the following axioms: For any x; y; z 2 X
C1: ((x y ) (z y )) (x z ) = 0;
C2: 0 x = 0;
C3: x 0 = x;
C4: x y = y x = 0 imply x = y .
Denition 1.1. An algebra (X ;
This denition is the dual form of that given by Y. Komori in [1]. The two denitions
are equivalent to each other. But we think that Denition 1.1 given here is more convenient
for us in comparing BCC-algebras with BCK-algebras.
From Denition 1.1 and the basic properties of BCK-algebras it is obvious that a BCKalgebra is a BCC-algebra. But we shall show later that a BCC-algebra need not be a
BCK-algebra. Therefore the concept of a BCC-algebra is a generalization of that of a
BCK-algebra.
Proposition 1.1. Let X = (X ;
; 0) be a BCC-algebra, then
C5: x x = 0; 8x 2 X
holds in X . (see Lemma 3 of [4])
Denition 1.2. Let X = (X ;
xy
() x y = 0:
; 0) be a BCC-algebra. For x; y 2 X , dene
" " is called the BCC-order on X .
AMS(1991) Subject classication. 06F35, 03G25.
Key words and phrases. BCK-algebra; BCC-algebra; ideal.
Project 19571022 supported by National Natural Science Foundation of China.
374
JIANG HAO
Proposition 1.2. The binary relation dened in Denition 1.2 is a partial ordering on
X . (cf.[5])
From C1, C2 and Denition 1.2 we obtain: In a BCC-algebra X , for any x; y; z 2 X the
following inequalities hold:
C1': (x y ) (z y ) x z ;
C2': 0 x.
We may call C1' the rule for cutting tail.
; 0) be a BCC-algebra, x; y; z 2 X , then
(i) x y =) x z y z:
This means that the right multiplication preserves the ordering.
(ii) x y =) z y z x:
This means that the left multiplication reverses the ordering.
Proposition 1.3. Let X = (X ;
Proof. (i) By C1' we have
(x z ) (y z ) x y
(1.1)
x y = 0:
(1.2)
From x y and Denition 1.2 it follows that
By (1.1) and (1.2) we have
(x z ) (y z ) 0:
This means that (x z ) (y z ) = 0, since 0 is the least element of X by C2'. Hence we
obtain x z y z:
(ii) By C1' we obtain
(z y ) (x y ) z x
(1.3)
And x y means
xy =0
By (1.3) and (1.4) we have
Using C3 we see that
(1.4)
(z y ) 0 z x:
z y z x:
Proposition 1.4. In a BCC-algebra X ,
x y x;
8x; y 2 X:
Proof. By C2' we have 0 y . Using Proposition 1.3 (ii) and C3 we obtain
xy
Proposition 1.5. Let X = (X ;
only if it satises:
x 0 = x: ; 0) be a BCC-algebra, then X is a BCK-algebra if and
(x y ) z = (x z ) y;
8x; y; z 2 X
(cf. Lemma 2 of [4])
We call condition (1.5) the commutative law with the xed head.
(1.5)
IDEAL THEORY OF BCC{ALGEBRAS
375
; 0) be a BCC-algebra and H a nonempty subset of X . If
H is itself a BCC-algebra under the operation in X , then H is called a subalgebra of X
and denoted by H X . The symbol H < X means that H X and H 6= X .
Denition 1.3. Let X = (X ;
Proposition 1.6. A nonempty subset H of a BCC-algebra X is a subalgebra if and only
if it is closed under the operation in X .
Proof. This is obvious by Denition 1.3.
X = (X ; ; 0) and X 0 = (X 0 ; 0 ; 00 ) be two BCC- algebras. A
0
mapping f : X ! X is called a homomorphism from X into X 0 if, for any x; y 2 X ,
Denition 1.4. Let
f (x y ) = f (x) f (y ):
(1.6)
If, in addition, f (X ) = X 0 , then f is called an epimorphism and X 0 is said to be a homomorphic image of X . A homomorphism f is called a monomorphism if it is one-to-one. f is
called an isomorphism if it is both an epimorphism and a monomorphism. We denote this
fact by X = X 0.
The phrase "f : X ! X 0 is a BCC-homomorphism" is often used to express the fact
that f is a homomorphism from the BCC-algebra X to the BCC-algebra X 0 .
When no confusion may arise, we often use the same symbol to denote the operations
both in X and inX 0 , even if they are in fact dierent. Our purpose in doing so is to simplify
the symbol. The zero elements of X and X 0 are also often denoted by the same symbol 0.
Proposition 1.7. Let f : X
! X 0 be a BCC-homomorphism, then
(i) f (0) = 0 ;
(ii) x y implies f (x) f (y ).
Proof. (i) Suppose that x is an element of X , then by C5 and (1.6) we have
f (0) = f (x x) = f (x) f (x) = 0:
(ii) x y Def.1.2
=) x y = 0 =) f (x y ) = 0 =) f (x) f (y ) = 0 Def.1.2
=) f (x) f (y ):
(i)
(1.6)
! X 0 be a BCC-homomorphism. Set Kerf = fx 2 X jf (x) =
0g, we call Kerf the kernel of f . Set Imf = f (X ) = ff (x)jx 2 X g, we call Imf the image
Denition 1.5. Let f : X
of f .
Proposition 1.8. Let
Imf X 0 .
f : X
!
X 0 be a BCC-homomorphism, then Kerf
Proof. The proof is routine and easy and so is left to the reader.
X and
x2. Relationship between BCC-algebras and other algebras.
Theorem 2.1. Let I be the class of BCI-algebras, K the class of BCK-algebras and C the
class of BCC-algebras respectively, then
K = I \ C:
Proof. Obviously, K I . From Denition 1.1 and the basic properties of BCK-algebras we
see that K C . Hence
K I \ C:
(2.1)
376
JIANG HAO
Conversely, if X 2 I \ C , then X is a BCC-algebra, and a BCI-algebra as well. X satises
condition (1.5), since it is a BCI-algebra. Therefore by Proposition 1.5 we have X 2 K. It
follows that
I \C K
(2.2)
From (2.1) and (2.2) we have K = I \ C :
For basic properties of BCH-algebras the reader is referred to [8].
Theorem 2.2. The meaning of the symbols I ; K; C is the same as in Theorem 2.1. Let H
be the class of BCH-algebras. Then
Proof. It is well known that I
K = H \ C:
H, hence by Theorem 2.1 we obtain
K =I \C H\C
(2.3)
If X 2 H \ C , then X is a BCC-algebra, and a BCH-algebra as well. By the denition of
a BCH-algebra (cf.[8]) we see that X satises condition (1.5). So X is a BCK-algebra by
Proposition 1.5. Thus we have
H\C K
(2.4)
From (2.3) and (2.4) it follows that K = H \ C :
Denition 2.1. If
X is a BCC-algebra but it is not a BCK- algebra, then we call it a
proper BCC-algebra.
By Theorem 2.1 and 2.2 we see that a proper BCC-algebra is neither a BCH- algebra
nor a BCI-algebra, let alone a BCK-algebra. In the next section we deal with the existence
of proper BCC-algebras.
x3. Existence of proper BCC-algebras.
Theorem 3.1. Let X = (X ; ; 0) be a proper BCC-algebra, then jX j 4:
Proof. Let X be a BCC-algebra. We prove that X is a BCK-algebra if jX j 3: By
Proposition 1.5 it suÆces to show that X satises condition (1.5).
Since jX j 3, for any x; y; z 2 X there are only two possibilities as follows:
Case 1. x; y; z are dierent to each other. In this case, one of them must equal 0.
(i) If x = 0, then we have (x y ) z = (0 y ) z C2
= 0 C2
= (0 z ) y = (x y ) z:
(ii) If y = 0, then
(x y ) z = (x 0) z C3
= x z:
(x z ) y = (x z ) 0 C3
= x z:
So (x y ) z = (x z ) y holds.
(iii) If z = 0, then we obtain
(x y ) z = (x y ) 0 C3
= x y;
(x z ) y = (x 0) y C3
= x y:
Hence we get (x y ) z = (x z ) y:
Case 2. At least two of x; y; z are equal.
(i) If x = y , then (x y ) z = (x x) z C5
= 0 z C2
= 0: By Proposition 1.4 we have
(x z ) y = (x z ) x = 0: So (x y ) z = (x z ) y:
IDEAL THEORY OF BCC{ALGEBRAS
377
(ii) If x = z; then by Proposition 1.4 it follows that (x y ) z = (x y ) x = 0: By C5
and C2 we have (x z ) y = (x x) y = 0 y = 0: Hence (x y ) z = (x z ) y:
(iii) If y = z , then (x y ) z = (x y ) y = (x z ) y:
Summarizing the preceding facts we see that X satises condition (1.5) if jX j 3. Hence
by Proposition 1.5 X is a BCK-algebra, and so it is not a proper BCC-algebra. j
0 1 2 3
j
j
j
j
0
0
0
0
0
1
2
3
0 0 0
1 0 0
2 1 0
3 2 2
Table 1(X1 )
j
0 1 2 3
j
0 1 2 3
j
j
j
j
0 0 0 0
1 0 0 1
2 1 0 1
3 3 3 0
Table 2(X2 )
0
1
2
3
j
j
j
j
0
1
1
0
0
1
2
3
0 0 0
1 0 0
2 2 0
3 3 3
Table 3(X3 )
By Table 1-Table 3 we give three examples of proper BCC-algebras of order four, denoted
by X1 , X2 , X3 respectively. Each contains a BCK-algebra of order three. X1 and X2 contain
B3 2 1 . X3 contains B3 1 1 . As for the meaning of the symbols B3 2 1 and B3 1 1 , the
reader is referred to [9].
x4. Ideal theory of BCC-algebras.
Denition 4.1. Let X = (X ; ; 0) be a BCC-algebra and be an equivalent relation on
X . For any x; y; u 2 X , if x y implies u x u y , then is called a left congruence. If
x y implies x u y u, then is called a right congruence.
Denition 4.2. Let X = (X ; ; 0) be a BCC-algebra and be an equivalent relation on
X . For any x; y; u; v 2 X , if x y; u v imply x u y v , then is called a congruence.
Proposition 4.1. Let X = (X ; ; 0) be a BCC-algebra and be an equivalent relation
on X . Then is a congruence if and only if it is both a left and a right congruence.
Proof. The proof is routine and easy, so is left to the reader. Denition 4.3. Let X = (X ; ; 0) be a BCC-algebra and ; =
6 I X . I is called an ideal
of X , denoted by I / X , if it satises the following conditions:
(i) 0 2 I ;
(ii) x y; y 2 I imply x 2 I .
We note that the conditions for an ideal in a BCC-algebra are the same as that for an
ideal in a BCI-algebra. Obviously X has two trivial ideals, i.e. f0g and X itself.
In BCC-algebra X2 as showed by Table 2, f0; 1; 2g and f0; 3g are non-trivial ideals. In
BCC-algebra X3 as showed by Table 3, we can see that f0; 1; 2g; f0; 1g and f0; 3g are
non-trivial ideals.
Theorem 4.1. In a BCC-algebra, an ideal is a subalgebra.
Proof. Let X be a BCC-algebra, I / X: For any x; y 2 I , by Proposition 1.4 we have
(x y ) x = 0 2 I . Since x 2 I and I is an ideal, it follows that x y 2 I . Therefore
I X. Denition 4.4. Let X be a BCC-algebra and a 2 X . Set A(a) = fx 2 X jx ag: We call
A(a) the initial section of the element a.
378
JIANG HAO
Theorem 4.2. Let X be a BCC-algebra, a 2 X , then A(a) X .
Proof. By C5 we have a a, so a 2 A(a): This shows that A(a) 6= ;: If x; y
a: From x a and Proposition 1.3(i) we obtain
x a; y
2 A(a), then
xy ay
It follows from Proposition 1.4 that
ay
Combining (4.1) and (4.2) we have x y
A(a) X:
a
a.
(4.2)
This means that x y
Theorem 4.3. Let X be a BCC-algebra. If I / X;
Proof. If y 2 A(x), then y
Therefore A(x) I: (4.1)
2 A(a): Therefore
x 2 I; then A(x) I .
x, hence y x = 0 2 I: Since x 2 X and I / X , so y 2 I .
Theorem 4.4. Suppose that I / X , where X is a BCC- algebra. Dene a binary relation
on X as follows: For any x; y
relation on X .
2 X , x y () x y; y x 2 I: Then is an equivalent
Proof. By Denition 4.3 it follows that 0 2 I , since I / X: For any x 2 X , using C5 we
have x x = 0 2 I , so x x:
If x y , then by the denition of it is obvious that y x:
Now assume x y; y z , then x y; y x 2 I and y z; z y 2 I: By C1 we have
(z x) (y x) z y 2 I . Using Theorem 4.3 it follows that (z x) (y x) 2 I: Since
y x 2 I and I / X , we have z x 2 I: In the same way we can prove x z 2 I . Therefore
x z: Summarizing the preceding facts we see that is an equivalent relation on X . on X described in Theorem 4.4 is called the
equivalent relation on X determined by I . The equivalent class containing x is denoted by
Ix , i.e. Ix = fy 2 X j y xg:
Denition 4.5. The equivalent relation
Theorem 4.5. The meaning of symbols is the same as in Dention 4.5. Let I
I0 = I .
Proof. x 2 I0 , x 0 , x 0; 0 x 2 I
/ X , then
, x 2 I: Theorem 4.6. Let X be a BCC-algebra, I / X: Then the equivalent relation
determined by I is a right congruence.
on X
Proof. Assume x y , then x y; y x 2 I . By C1' we have (x u) (y u) x y 2 I and
(y u) (x u) y x 2 I: Using Theorem 4.3 we obtain
and
(x u) (y u) 2 I
(4.3)
(y u) (x u) 2 I
(4.4)
From (4.3) and (4.4) we have x u y u. This means that is a right congruence.
; 0) be a BCC-algebra, I / X: We call I a regular ideal if
the equivalent relation on X determined by I is a congruence.
Denition 4.6. Let X = (X ;
Obviously f0g and X are two trivial regular ideals of X .
IDEAL THEORY OF BCC{ALGEBRAS
379
Theorem 4.7. Let
X be a BCC-algebra and I / X , then I is regular if and only if the
equivalent relation on X determined by I is a left congruence.
Proof. Ths is the direct consequence of Proposition 4.1, Theorem 4.6 and Denition 4.6. Now let us observe the proper BCC-algebra X2 given by Table 2. By Theorem 4.7 it
is easy to verify that in this algebra I = f0; 1; 2g is a regular ideal and J = f0; 3g is a
non-regular ideal.
In the proper BCC-algebra X3 given by Table 3 we can see that I = f0; 1; 2g and
K = f0; 1g are regular ideals, but J = f0; 3g is a non-regular ideal.
Theorem 4.8. Let X = (X ; ; 0) be a BCC-algebra and I be a regular ideal of X . Dene
a binary operation on the quotient set X=I = fIx j x 2 X g such that Ix Iy = Ixy ; then
(X=I ; ; I ) is a BCC-algebra, called the quotient algebra of X relative to I .
Proof. Since I is regular, the equivalent relation on X determined by I is a congruence.
If Ix = Ix ; Iy = Iy ; then x0 x; y 0 y . It follows that x0 y 0 x y; since is a
congruence. Hence Ix Iy = Ix y = Ixy = Ix Iy . This shows that the binary operation
on X=I is well dened. Now it is routine and easy to verify that C1, C2, C3 and C4 hold
in X=I: Therefore (X=I ; ; I ) is a BCC-algebra. Theorem 4.9 (Fundamental Homomorphism Theorem). Let f : X ! X 0 be a
BCC-epimorphism, K = Kerf , then
(i) K is a regular ideal of X ;
(ii) X=K = X 0:
0
0
0
0
0
0
Proof. (i) By Proposition 1.7(i) we have 0 2 K: Assume that x y; y 2 K , then f (x y ) = 0; f (y ) = 0: It follows that f (x) f (y ) = 0; f (y ) = 0: Now it is easy seen that
f (x) = f (x) 0 = 0, hence x 2 K . Thus we have showed that K is an ideal of X .
Let be the equivalent relation on X determined by K . For x; y; u 2 X , suppose that
x y . Then x y; y x 2 K . It follows that f (x y ) = 0; f (y x) = 0 and hence
f (x) f (y ) = 0; f (y ) f (x) = 0. By C4 we have f (x) = f (y ): Left -multiplying both sides
of this equation by f (u) we have f (u)f (x) = f (u)f (y ): This implies that f (ux) = f (uy )
since f is a homomorphism. By C5 it follows that f (u x) f (u y ) = 0; f (u y ) f (u x) = 0,
hence we have f ((ux)(uy )) = 0; f ((uy )(ux)) = 0: This means that (ux)(uy ) 2 K
and (u y ) (u x) 2 K , so u x u y: Thus we have proved that is a left congruence.
By Theorem 4.7 we see that K is a regular ideal of X .
(ii) Since K is a regular ideal of X , by Theorem 4.8 we obtain the quotient algebra
X=K . Assume f : X=K ! X 0 such that f(Kx ) = f (x). In the following we prove
that f is an isomorphism. If Kx = Ky , then x y , so x y; y x 2 K: It follows that
f (x y ) = 0; f (y x) = 0: This imply f (x) f (y ) = 0; f (y ) f (x) = 0. By C4 we have
f (x) = f (y ), i.e., f(Kx ) = f(Ky ). This shows that f is a well dened mapping from
X=K to X 0 . For any x0 2 X 0 , there exists x 2 X such that x0 = f (x) as f is onto.
Hence f(Kx ) = f (x) = x0 , which means that f is onto. If Kx 6= Ky , then x and y do
not belong to the same equivalent class. Thus x y 2= Kerf or y x 2= Kerf . Suppose
that x y 2= Kerf without loss of any generality, it follows that f (x) f (y ) = f (x y ) 6=
0. Therefore f (x) 6= f (y ), i.e. f(Kx ) 6= f(Ky ). This says that f is one-to-one. Since
f(Kx Ky ) = f(Kxy ) = f (x y ) = f (x) f (y ) = f(Kx ) f(Ky ), f is a homomorphism.
Putting the above facts together we know that f is an isomorphism from X=K onto X 0 .
Therefore X=K = X 0: In [7] we introduced the concept of a BCK-algebra with a small atom. Now we show
that every BCK-algebra with a small atom can be extended to a proper BCC- algebra by
adding a maximal element.
380
JIANG HAO
Let X = (X ; ; 0) be a BCK-algebra and a 2 X . If a satises (i) a 6= 0; (ii)a x =
0; 80 6= x 2 X , then it is called a small atom of X . (see Denition 1 of [7].)
X = (X ; ; 0) be a BCK-algebra with the small atom a such that
Assume that u 2
= X and set Y = X [ fug. Dene a binary operation 0 on Y as
Theorem 4.10. Let
jX j > 2.
follows:
8 x y;
>
>
>
< u;
x 0 y = 0;
>
>
>
: a;0;
if
if
if
if
if
x; y 2 X
x = u; y 2 X
x = 0; y = u
x = u; y = u
x 2 X f0g; y = u
(4.5)
(4:6)
(4:7)
(4:8)
(4:9)
then (Y ; 0 ; 0) is a proper BCC-algebra, u is a maximal element of Y , and X is a regular
ideal of Y .
Proof. It is easily seen that C2, C3, and C4 hold in Y . Obviously C5 also holds in Y . To
verify C1, we consider the following possible cases :
Case 1: If x; y; z 2 X , then C1 holds, since X is a BCK-algebra.
Case 2: If one elenent of x; y and z is equal to zero, it is easily seen that C1 holds.
Case 3: If two elements of x; y and z are equal, it is also very easy to verify that C1
holds.
Therefore we now need only to consider the cases in which x; y; z are dierent to each
other and none of them is equal to zero. There are the following possibilities:
Case 4: y; z 2 X and x = u.
(4.5), (4.6)
(4.6)
Then ((x 0 y ) 0 (z 0 y )) 0 (x 0 z ) = ((u 0 y ) 0 (z 0 y )) 0 (u 0 z ) === (u 0 (z y )) 0 u =
(4.8)
u 0 u = 0:
Case 5: x; z 2 X and y = u.
In this case we have
(4.5)
(4.9)
(4.5)
((x 0 y ) 0 (z 0 y )) 0 (x 0 z ) = ((x 0 u) 0 (z 0 u)) 0 (x z ) = (a 0 a) 0 (x z ) =
(a a) (x z ) = 0 (x z ) = 0: (Note that X is a BCK-algebra.)
Case 6: x; y 2 X and z = u:
It follows that
(4.5)
(4.6),(4.9)
((x 0 y ) 0 (z 0 y )) 0 (x 0 z ) = ((x y ) 0 (u 0 y )) 0 (x 0 u) === ((x y ) 0 u) 0 a = 0,
since (x y ) 0 u = 0 or a.
Summarizing the preceding facts we see that C1 holds in Y . Hence by Denition 1.1
(Y ; 0 ; 0) is a BCC-algebra.
Now we show that (Y ; 0 ; 0) is a proper BCC-algebra.
Since jX j > 2, we have X f0; ag 6= ;. So there exists x 2 X f0; ag. It follows that
a < x, since a is the small atom of X . Hence a 0 x = a x = 0 and x a 2 X f0g. From
(4.5)
(4.9)
(4.9)
(4.5)
these facts we obtain (x 0 a) 0 u = (x a) 0 u = a: But (x 0 u) 0 a = a 0 a = a a = 0:
Then by Proposition 1.5 and Denition 2.1 we see that (Y ; 0 ; 0) is a proper BCC- algebra.
Finally we prove that X is a regular ideal of Y .
It is obvious that 0 2 X: Assume that x 0 y 2 X; y 2 X: If x 2= X , then x 2 Y X = fug,
(4.6)
i.e. x = u: Hence u = u 0 y = x 0 y 2 X , a contradiction. Therefore we must have x 2 X ,
by Denition 4.3 X is an ideal of Y . Let be the equivalent relation on Y determined by
X , then there are only two equivalent classes: X and fug.
Suppose that x; y 2 Y = X [ fug and x y , then there are only two possibilities as
follows:
IDEAL THEORY OF BCC{ALGEBRAS
381
(i) x; y 2 X .
In this case, for any z 2 X we have
z 0 x = z x 2 X;
(4.5)
z 0 y = z y
(4.5)
2 X;
so z 0 x z 0 y .
If z = u, then by (4.6) it follows that u 0 x = u; u 0 y = u; hence u 0 x u 0 y .
(ii) x = y = u:
In this case, u u: So for any z 2 Y , we have z 0 u x 0 u.
From (i) and (ii) we see that is a left congruence, hence by Theorem 4.7 X is a regular
ideal of Y . References
[1] Y. Komori, The class of BCC-algebras is not a variety, Math. Japonica, 29(1984),
391{394.
[2] K.Iseki and S. Tanaka, An introduction to the theory of BCK-algebras, Math.
Japonica, 23(1978), 1{26.
[3] J. Meng and Y.B. Jun, BCK-algebras, Kyung Moon SA Co., Seoul, Korea, 1994.
[4] Dong Liezhao and Jiang Hao, On the duality between BCC- algebras and BCKalgebras, J. of Nanchang Univ. (Nat. Sci. Edi.), 1995; Vol.19(Supplement), 28{30.
[5] W.A. Dudek, On constructions of BCC-algebras, Selected Papers on BCK-and
BCI-algebras, Vol.1,(1992), 93{96. Shensi Scientic and Technological Press, Xi'an
China.
[6] Y. Komori, The variety generated by BCC-algebras is nitely based, Reports Pac.
Sci. Shizuoka Univ., 17(1983), 13{16.
[7] Jiang Hao, On BCK-algebras with a small atom, Math. Japonica, 44(1996), 357{
362.
[8] Hu Qingping and Li Xin, On BCH-algebras, Math. Sem. Notes (Kobe Univ.),
11(1983),313{320.
[9] Jiang Hao, Computational methods in the study of nite BCK-algebras with low
orders, Kobe J. Math., 7(1990), 33{46.
Department of Mathematics, Hangzhou University, Hangzhou 310028, P.R. China
E-mail :
[email protected]