Filters in pseudo-BCK algebras [611999]

Filters in pseudo-BCK algebras
Mihaela Istr at a
Faculty of Sciences, Department of Mathematics, University of Craiova,
Romania
Abstract
An important concept in the theory of residuated lattices and other algebraic struc-
tures used for formal fuzzy logic is that of a lter. In this paper, we study the properties
and relations of pseudo-lters of pseudo-BCK algebras and after we dene a new type
of lters its called crazy-pseudo-BCK lters and complet-pseudo-BCK lters.
1 Introduction
The logical algebras are the algebraic counterpart of the non-classical logic. BCK-algebras,
BL-algebras, pseudo-MTL algebras and non-commutative residuated lattice are algebraic
countpart of BCK Logic, Basic Logic, monoidal t-norm-based logic and monoidal logic re-
spectively.
Iseki and Imai introduced BCK-algebra for BCK Logic in ([8]). Afterwards, Georgescu and
Iorgulescu introduced notion of pseudo-BCK algebra as extension of BCK algebra ([4], [6]).
Iorgulescu estabilished connections between pseudo-BCK algebra and pseudo-BL algebra.
Filters are subsets of partially ordered sets that satisfy some denig properties.The con-
cept is also used in partially ordered algebraic structured with more connectives (lattices,
residuated lattices), in particular in the algebraic semantics of formal fuzzy logics. Filter
theory plays an important role both in algebraic structure research and non-classical logic
and computer science. From logical point of view, various lter corresponds to various set
of provable formulae. Based on lters and prime lters in BL-algebras, H ajek proved the
completenes of Basic Logic BL.
2 Preliminaries
In this section we recall some basic results about pseudo-BCK algebras, pseudo-BCK lters.
We recall that a pseudo-BCK algebra is a structure ( A;;!; ;1) whereis a binary
relation on A!and are binary operations on A and 1 is an element of A satisfying for
allx; y; z2Athe axioms:
(a1)x!y(y!z) (x!z), (x y)(y z)!(x z),
(a2)x(x!y) y,x(x y)!y,
1

(a3)xx,
(a4)x1,
(a5) ifxyandyxthen x=y,
(a6)xyix!y= 1 i x y= 1.
We remark that a pseudo-BCK algebra is commutative if != .
Any commutative pseudo-BCK algebra is a BCK algebra.
In any pseudo-BCK algebra the following properties hold ([2], [3]):
(c1)xyimplies y!zx!zandy zx z;
(c2)xy; yzimpilies xz;
(c3)x!(y z) =y (x!z) and x (y!z) =y!(x z);
(c4)zy!xiyz x;
(c5)z!x(y!z)!(y!x); z x(y z) (y x);
(c6)xy!x; xy x;
(c7) 1!x=x= 1 x;
(c8)xyimplies z!xz!yandz xz y;
(c9) [(y!x) x]!x=y!x;[(y x)!x] x=y x.
Denition 1. ([1]) If there is an element 0 of a pseudo-BCK algebra (A;;!; ;1)such
that 0x(i.e. 0!x= 0 x= 1) for all x2A, then it is called the zero of A. A
pseudo-BCK algebra with zero is called a bounded pseudo BCK algebra and is denoted by
(A;;!; ;0;1).
For all x2A, a bounded pseudo-BCK algebra, we have two negations: x=x!0; x=
x 0.
Proposition 1. ([6],[3]) In any bounded pseudo-BCK algebra, for all x2A,0_x=x= 0[x
andx_0 = (x); x[0 = (x).
Proposition 2. In any pseudo-BCK algebra A the following properties hold for all, x; y2A:
(i) 1_x=x_1 = 1 = 1[x=x[1;
(ii)xyimplies x_y=yandx[y=y;
(iii)x_x=x[x=x;
Proof. (i) We have 1_x= (1!x) x= 1 and x_1 = ( x!1) 1 = 1, so
1_x=x_1 = 1. Similarity, 1 [x=x[1 = 1.
(ii)x_y= (x!y) y= 1 y=y. Similarity, x[y=y.
(iii) By denition of _and[.
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Proposition 3. In any pseudo-BCK algebra, for all x; y2A,x_y!y=x!yand
x[y y=x y.
A pseudo-BCK algebra ( A;;!; ;1) in which the poset ( A;) is a lattice is called a
poset BCK lattice and a pseudo BCK (pP) lattice is a pseudo BCK lattice satisfying the
pseudo condition: For all x; y; z2A,xyexists, where xy=minfz:zy!zg=
minfz:yx zg.
In any pseudo-BCK algebra (pP) the following properties hold ([3]):
(c0
1)xyx; y;
(c0
2) (x!y)xx; x(x y)x; y;
(c0
3)yx!(yx); yx (xy);
(c0
4)x!y(xz)!(yz); x y(zx) (zy);
(c0
5)x(y!z)y!(xz);(y z)xy (zx);
(c0
6) (y!z)(x!y)x!z;(x y)(y z)x z;
(c0
7)x!(y!z) = (xy)!z; x (y z) = (yx) z;
(c0
8) (xz)!(yz)x!(z!y);(zx) (zy)x (z y);
(c0
19)x!y(xz)!(yz)x!(z!y); x y(zx) (zy)x (z y);
(c0
10)xyimplies xzyz;zxzy.
Proposition 4. In a pseudo-BCK algebra A, for all x; y; z2Athe followings hold
(i)x_y= ((x!y) y)^((y!x) x);
(ii)x_y= ((x y)!y)^((y x)!x);
(iii) x!y=x!x^y; x y=x x^y.
Denition 2. In a pseudo-BCK algebra, if an element x satises relations: x=x=x,
then x is called an involution. If every element of A is an involution, we say that A is an
involutory pseudo-BCK algebra.
In a bounded pseudo-BCK algebra, we have the following properties:
(c10) 1= 0 = 1;0= 1 = 0;
(c11)x(x); x(x);
(c12)x!yy x; x yy!x;
(c13)xyimplies yxandyx;
(c14)x!y=y xandx y=y!x;
(c15) ((x))=x;((x))=x.
3

3 Pseudo-lters of pseudo-BCK algebra
In this section, we recall the denitions of pseudo lter and fuzzy lter in pseudo-BCK
algebra A.
Denition 3. A nonempty subset F of a pseudo-BCK algebra A is called a pseudo-lter if:
(i)x2F; y2A; xy)y2F,
(ii)x2F; x!y2Forx y2F)y2F.
Theorem 1. ([9]) A subset F of A is a pseudo-lter if and only if
(i)12F,
(ii)x2F,x!y2Forx y2Fimplies y2F.
Theorem 2. Let be F a nonempty subset F of a pseudo-BCK algebra A with (pP) condition.
Then F is a pseudo-lter if and only if
(i)x; y2Fimplies xy2F;
(ii)x2F; xy2Aimplies y2F.
Proof. If F is a pseudo-lter of A and x; y2F, then x!(y!(xy)) = ( xy)!
(xy) = 12Fandy!(xy)2F, so we have xy2F.
Conversely, if F satises (i) and (ii), then x;(x!y) or ( x y)2F, so ( x!y)x,
x(x y)y2F.
Denition 4. A pseudo-lter F of a pseudo-BCK algebra A is normal, if x!y2Fi
x y2F.
Denition 5. ([10])For x; y2A, a lter F is called
(i)a Boolean if (x!y) x2Fand(x y)!x2F;
(ii) an implicative if (x!y)!x2Fand(x y) x2F, then x2F.
Theorem 3. ([10]) Let F be implicative of a bounded pseudo-BCK algebra A. Then for all
x; y2Awe have:
(i)((x!0)!x) x2F;((x 0) x)!x2F;
(ii)((x!y)!x) x2F;((x y) x)!x2F;
(iii) if(x!y)!y2F, then (y!x) x2F, if(x y) y2F, then (y x)!
x2F;
(iv) ifx y2F, then ((y x)!x) y2F, ifx!y2F, then ((y!x) x)!
y2F.
4

Denition 6. Let be F a nonempty subset of a pseudo-BCK algebra A. Then F is a postive
implicative pseudo-lter of A if for all x; y2Athe following hold:
(i)x2F; y2A; xyimplies y2F,
(ii)x (y!z); x y2Fimplies x z2F,
(iii) x!(y z); x!y2Fimplies x!z2F.
Denition 7. ([9]) A fuzzy subset f of A is a fuzzy pseudo-lter if ftis either empty or a
pseudo-lter for t2[0;1].
F is a pseudo-lter i XFis a fuzzy pseudo-lter, where XFis the characteristic function
of F.
Proposition 5. A fuzzy set f is a fuzzy pseudo-lter of A (or A(pP)) ([9]) if and only if for
x; y; z2A, one the followings holds:
(i)f(1)f(x); f(y)f(x)^f(x!y); f(1)f(x); f(y)f(x)^f(x y);
(ii) is order-preserving and f(xy)f(x)^f(y).
Denition 8. Forx; y2A, a fuzzy pseudo-lter f is called:
(i)implicative if f((x!y)!x) =f(x); f((x y) x) =f(x);
(ii) Boolean if f((x!y) x) =f((x y)!x) =f(x).
Theorem 4. ([9]) If f is a fuzzy implicative pseudo-lter of a bounded pseudo-BCK algebra
A then8x2A; f((x!x) x) =f((x x)!x) =f((x x)!x) =f((x!
x) x) =f(1).
Proof. From x(x!x) x, so (( x!x) x)and (( x!x) x)!x= 1,
thenf(((x!x) x)) =f(1). And x(x!x) x, so we get (( x!x) x)!
x((x!x) x)!((x!x) x).
Then we prove that f(((x!x) x)!((x!x) x)) =f((x!x) x) = 1.
Similary we can show that f((x x)!x) =f(1).
From x(x x)!x, so (( x x)!x)xand (( x x)!x)!
x= 1, then f(((x x)!x)!x) =f(1). And x(x x)!x, so we
get (( x x)!x)!x((x x)!x)!((x x)!x).Thus we prove
f(((x x)!x)!((x x)!x)) =f((x x)!x) =f(1).
Similary f((x!x) x) =f(1).
Theorem 5. ([9]) A fuzzy pseudo-lter F of a pseudo-BCK algebra A is implicative if and
only if ftis either empty or an implicative pseudo-lter for each t2[0;1]
Proof. Let F be a fuzzy implicative pseudo-lter of A. For any t2[0;1], if ft6= , then
suppose ( x!y)!x2ft, then from f((x!y)!x) =f(x), we get x2ft. Dually we
can get x2ftif (x y) x2ft. Soffis an implicative pseudo-lter of A.
Conversely, let f((x!y)!x) =t, we get ( x!y)!x2ft, and ftis an implicative
pseudo-lter of A, then x2ft. That is, f((x!y)!x) =tf(x). Since x(x!y)!x
and f is isotone, then f((x!y)!x)f(x), we get f((x!y)!x) =f(x). Dually we
can obtain f((x y) x) =f(x), thus proves that f is implicative.
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Theorem 6. ([9]) A fuzzy pseudo-lter f of a bounded A is Boolean if and only if f(x!
y) =f(x!(y y)).f(x y) =f(x (y!y))if and only if f((x!x) x) =
f((x x)!x) =f(1)forx; y2A.
Theorem 7. ([9]) If f is a fuzzy Boolean lter of a bounded pseudo-BCK algebra A, then
f((y x)!x) =f((x!y) y)forx; y2A.
In every f fuzzy Boolean lter of a bounded A we have f(x) =f(x) =f(x) for
x2A.
Denition 9. A nonempty subset F of bounded pseudo-BCK algebra A is called a crazy
lter if :
(i)12F;
(ii)(x!y)2For(x y)2F,y2Fimplies x2F.
Let A, B be two pseudo-BCK algebras. A map f:A!Bis called a pseudo-BCK
homomorphism if f(x!y) =f(x)!f(y) and f(x y) =f(x) f(y), for all x; y2A.
IfA=B, then f is called a pseudo-BCK endomorphism.
If f is a homomorphism and onto, then f is a epimorphism.
If f is homomorphism and one to one then f is monomorphism.
If f is epimorphism and monomorphism then f is isomorphism.
Lemma 1. If f is a homomorphism from pseudo-BCK algebra A into pseudo-BCK algebra
B, then f is isotone i.e.:
(i)f(1) = 1 ;
(ii)xyimplies f(x)f(y), for all x; y2A;
If f is a bounded pseudo-BCK homomorphism such that f(0) = 0 , then the following
hold:
(iii) f(x) =f(x);
(iv) f(x) =f(x).
Lemma 2. Let f be an isomorphism from pseudo-BCK algebra A into pseudo-BCK algebra
B, then
(i)f(x) =f(x),f(x) =f(x)
(ii)f1(y) = (f1(y)),f1(y) = (f1(y)).
Proof. (i)f(x) =f(x!0) = f(x)!f(0) = f(x)!0 =f(x),f(x) =f(x 0) =
f(x) f(0) = f(x) 0 =f(x).
(ii)f1(y) =f1(y!0) = f1(y)!f1(0) = f1(y)!0 = ( f1(y)),f1(y) =
f1(y 0) =f1(y) f1(0) = f1(y) 0 = (f1(y)).
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Theorem 8. If f is an isomorphism from A to B, two pseudo-BCK algebras, then the image
of a crazy lter is a crazy lter.
Proof. Let f be an isomorphism from A to B, two pseudo-BCK algebras and X be a crazy
lter in A, then 1 A2Xsof(1A) = 1 B2f(X) (Lemma1).
Let (x!y)2f(X),y2f(X), then f1((x!y))2X,f1(y)2X. But ( f1((x!
y)) = (f1(x!y))= (f1(x)!f1(y))by (Lemma2).
Therefore, ( f1(x)!f1(y))2X,f1(y)2X.
Now let ( x y)2f(X),y2f(X), then f1((x y))2X,f1(y)2X. But
(f1((x y)) = (f1(x y))= (f1(x) f1(y))by (Lemma2).
Therefore, ( f1(x) f1(y))2X,f1(y)2X.
Since X is a crazy lter in A, then f1(x)2X, thus x2f(X), that means f(X) is a
crazy lter in B.
Theorem 9. If f is an epimorphism from A into B, two pseudo-BCK algebras, then the
inverse image of a crazy lter is a crazy lter.
Proof. Let f be a epimorphism from pseudo-BCK algebra A into pseudo-BCK algebra B
and let be Y a crazy lter in B. Then 1 B2Ysince f(1A) = 1 B2Y, (by Lemma1), thus
1A2f1(Y).
Let (x!y)2f1(Y); y2f1(Y), sof((x!y))2Y; f(y)2Y. But f((x!y)) =
(f(x!y))= (f(x)!f(y)), (by Lemma2). Therefore ( f(x)!f(y))2Y; f(y)2Y,
thenf(x)2Y, since Y is a crazy lter, thus x2f1(Y), that means f1(Y) is a crazy lter
in A.
Now let ( x y)2f1(Y); y2f1(Y), so f((x y))2Y; f(y)2Y. But f((x
y)) = ( f(x y))= (f(x) f(y)), (by Lemma2). Therefore ( f(x) f(y))2
Y; f(y)2Y, then f(x)2Y, since Y is a crazy lter, thus x2f1(Y), that means f1(Y)
is a crazy lter in A.
Denition 10. A subset F af a bounded pseudo-BCK algebra A is complete pseudo-BCK
ltre if:
(i)12F;
(ii)(x!y)2For(x y)2F,8y2Fimplies x2F.
Example 1. LetA=f0; a; b; 1gwith 0< a; b < c < 1anda; bincomparable. Consider the
operation!; given by tables:
! 0 a b 1
0 1 1 1 1
a a 1 1 1
b b b 1 1
1 0 b b 1
0 a b 1
0 1 1 1 1
a a 1 1 1
b b b 1 1
1 0 0 a 1
7

Then ( A;;!; ;0;1) is a bounded pseudo-BCK algebra with 0 < a < b < 1 and
F=fa;1gis a complet pseudo-BCK lter.
Proposition 6. In a bounded pseudo-BCK algebra A, every pseudo-BCK lter is a complete-
pseudo-BCK lter.
Proof. Let F be a pseudo-BCK lter and ( x!y)2For (x y)2F. If F is a
pseudo-BCK lter, then x2F. Thus F is a complete-pseudo-BCK lter.
In generalf1g, F are trivial complete-pseudo-BCK-lter.
Proposition 7. Let f be isomorphism from a pseudo-BCK algebra A into a bounded pseudo-
BCK algebra B, the image of a complete-pseudo-BCK lter is a complete-pesudo-BCK lter.
Proof. Let f be an isomorphism function from a bounded pseudo-BCK algebra A into
bounded pseudo-BCK algebra B and let X be a complete-pseudo-BCK lter in A.
Then 1 A2Xso 1 A= 1 B2f(X). Let ( x!y)2Xor (x y)2X, then
f1((x!y))2X,f1((x y))2X,8f1(y)2X.
Butf1(x!y) = (( f1(x)))!((f1(y)))andf1(x y) = (( f1(x)))
((f1(y))). Therefore (( f1(x))!((f1(y)));((f1(x)) ((f1(y)))2F;8f1(y)2
F.
Since X is a complete-pseudo-BCK lter in A, then f1(x)2X. Thus x2f(X). So
f(X) is a complete-pseudo-BCK lter in B.
Proposition 8. Let f be epimorphism from a bounded pseudo-BCK algebra A into a bounded
pseudo-BCK algebra B, the inverse image of complete-pseudo-BCK lter is a complete-
pseudo-BCK lter.
Proof. Now let f be epimorphism from a pseudo-BCK algebra A into a bounded pseudo-BCK
algebra B and let Y be a complete-pseudo-BCK lter in B, so 1 B2Y.
Then f(1A) = 1 B2Yso 1 A2f1(Y). Let be ( x!y)2f1(Y)or(x y)2
f1(Y);8y2f1(Y), so f((x!y))2Y,f((x y))2Y,8f(y)2Y, since f is
onto. But f((x!y))=f((f(x)!(f(y))))2Y,f((x y)) =f((f(x))
(f(y)))2Y8f(y)2Y.Then f1(X) is a complete-pseudo-BCK lter.
Denition 11. A subset F of a pseudo-BCK algebra A is said to be complete crazy-lter, if
(i)12F;
(ii)(x!y)2For(x y)2F, for all y2Fimplies x2F.
Example 2. LetA=f0; a; b; c; 1gwith 0< a; b < c < 1anda; bincomparable. Consider
the operation!; given by tables:
! 0 a b c 1
0 1 1 1 1 1
a 0 1 1 1 1
b 0 a 1 c 1
c 0 b b 1 1
1 0 a b c 1
8

0 a b c 1
0 1 1 1 1 1
a 0 1 1 1 1
b 0 c 1 c 1
c 0 a b 1 1
1 0 a b c 1
It is clear ( A;!;1) is o pseuodo-BCK bounded algebra and F=f0;1gis complete-crazy-
lter.
4 Conection between the dierent kinds of pseudo-
lters of pseudo-BCK algebras
In ([11]) it is proved that in pseudo-BCK algebra or bounded pseudo-BCK algebra, implica-
tive pseudo lter is equivalent to Boolean lter.
Proposition 9. In a pseudo-BCK algebra (pP) every fuzzy Boolean lter is a fuzzy normal
lter.
Proposition 10. Every fuzzy Boolean lter of a pseudo-BCK algebra (pP) is implicative.
Proof. If f is a fuzzy Boolean lter we have that for any x; y2A; f((x!y) x) =
f(x); f((x y)!x) =f(x). So f((x!y)!x) =f(x); f((x y) x) =f(x). Thus F
is implicative.
Theorem 10. ([9]) A fuzzy pseudo-lter f of a pseudo-BCK algebra(pP) is a fuzzy normal
pseudo-lter if and only if for any x; y2A,f((xy) (yx))^f((yx)!(xy))f(y).
Proof. Let f be a fuzzy normal pseudo-lter. For any y2A, lett=f(y), then y2ftand
ftis a normal pseudo-lter of A. For any x2A, (yx)y=y(xy)xy, since
y(yx) (xy), then ( yx) (x y)2ftand ( yx)!(xy)2ft, thus
f((yx)!(xy))t=f(y). Dually, we have f((xy) (yx))t=f(y). From
above, then we obtain f((xy) (yx))^f((yx)!(xy))f(y) for any x; y2A.
Conversely, for any t2[0;1], if ft6= , then ftis a pseudo-lter and there exists y2ft.
f((yx)!(xy))y= (y!(x!(xy))yx!(xy), then we have
x!(xy)2ft, for any x2A; y2ft. For any a; b2A, ifa b2ft, from above, we get
a!a(a b) =a!b2ft. Dually, we can obtain x (yx)2ftfor any x2A; y2ft.
For any a; b2A, ifa!b2ft, from above, we can obtain a (a!b)a=a b2ft.
Thus ftis a normal pseudo-lter and f is normal.
Corollary 1. Every fuzzy Boolean lter of pseudo-BCK algebras (pP) is a fuzzy normal
lter.
Corollary 2. Every fuzzy Boolean lter of a pseudo-BCK algebra(pP) is implicative.
Corollary 3. If A is a pseudo-BCK algebra (pP) and f a fuzzy normal pseudo-lter, then
we can say that f is Boolean if and only if it is implicative.
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Theorem 11. ([10]) Let F be normal lter of A. Then F is implicative if and only if F is
Boolean.
Theorem 12. ([11])
If f is a fuzzy set of a pseudo-BCK algebra A, then the followings are equivalent for any
x; y; z2A.
(i)f is normal;
(ii) f is a fuzzy pseudo-lter satisfying f(x!y) =f(x y);
(iii) f(1)f(x); f(x z)f(y)^f(x!(y z))andf(x!z)f(y)^f(x (y!z);
(iv) f(1)f(x); f(x!z)f(y)^f(x!(y!z))andf(x z)f(y)^f(x (y
z)).
Proof. (i) implies ( ii). Let f(x!y) =t1, forx; y2A, sox!y2ft1, then x y2ft1,
f(x y)t1=f(x!y). Similary we can prove that f(x!y)f(x y).
(ii) implies ( iii). It is obvious.
(ii) implies ( iv). It is obvious.
(iii) implies ( i). If x= 1, we can obtain f is a fuzzy pseudo-lter. For any t2[0;1],
ifft6=  and x!y2ft, sof(x!y)t.x!((x!y) y) = 12ft, then
f(x!((x!y) y)) =f(1)tandf(x y)f(x!y)^f(x!((x!y) y))t,
sox y2ft. Also, we can prove that if x y2ft, then x!y2ft. Softis a normal
pseudo-lter and f is normal.
(iv) implies ( i). Similary of ( iii) implies ( i).
Theorem 13. ([9]) In bounded pseudo BCK algebra, every implicative pseudo lter is a
Boolean lter. In pseudo-BCK algebra (pP), every Boolean lter is an implicative pseudo-
lter.
Theorem 14. ([9])Let A be a pseudo-BCK algebra and F a normal pseudo-lter of A. Then
F is implicative if and only if F is Boolean.
Corollary 4. Fuzzy implicative psudo lters are fuzzy Boolean lters in pseudo BCK alge-
bras.
Theorem 15. ([11]) In pseudo-BCK algebras every fuzzy implicative pseudo-lters are fuzzy
Boolean lters.
Proof. Let f be an implicative pseudo-lter of A. Then 8x2A, suppose f((x!y) x) =t,
then ( x!y) x2ft. From x((x!y) x)!x), so ((( x!y) x)!x)!y
x!yand (((( x!y) x)!x)!y)!(x!y) = 1. But, x!y((x!y) x)!x,
so get (((( x!y) x)!x)!y)!(x!y)((((x!y) x)!x)!y)!(((x!
y) x)!x). Then (((( x!y) x)!x)!y)!(((x!y) x)!x) = 12ftand
((x!y) x)!x2ftsince f is an fuzzy implicative lter. Combine that ( x!y) x2
ft, according to the denition of lter, we get x2ft.
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Similarly, suppose f((x y)!x) =t, then ( x y)!x2ft. From x((x
y)!x) x, so ( x y)((x y)!x) x, so we get ((( x y)!x)
x) y) (x y)((((x y)!x) x) y) (((x y)!x) x). Then
((((x y)!x) x) y) (((x y)!x) x) = 12ftand (( x y)!x) x2ft
since f is a fuzzy implicative lter. Combine that ( x y)!x2ft, according to the
denition of lter, then we get x2ft. So, f is fuzzy Boolean lter.
Similary it is proved that in pseudo-BCK algebras, every fuzzy Boolean pseudo lter is a
fuzzy implicative lter.
Theorem 16. ([11])If A is a pseudo-BCK algebra or bounded pseudo-BCK algebra, implica-
tive lter and Boolean lter are equivalent.
The equivalence relation between the implicative pseudo lter and Boolean lter is impor-
tant because with that it is prove that a pseudo-BCK algebra is an implicative pseudo-BCK
algebra if and only if every pseudo-lter of it is a Boolean lter.
Theorem 17. ([10])A pseudo-BCK algebra (A;;!; ;0;1)is a bounded implicative BCK
-algebra if and only if every pseudo-lter is Boolean lter.
Theorem 18. . Let A be a pseudo-BCK algebras (pP). Then we can say that A is implicative
BCK algebra if and only if every pseudo-lters of them is Boolean lter.
Proof. If A pseudo-BCK algebras is implicative we know that x!y=x y, for all
x; y2Aand every pseudo-lters of them is implicative pseudo-lters.
Let be every pseudo-lter F of A a Boolen lter. Then every fuzzy pseudo-lter XFof
A is a fuzzy Boolean lter and XFis normal. Thus F is a positive implicative pseudo-lter
and ( A;!;1) is a positive implicative BCK-algebra and for all x; y2A,x!y=x y.
So all pseudo-lters of A is positive implicative.
Corollary 5. ([12]) Pseudo-BCK algebras (pP) is implicative BCK-algebras if and only if
every pseudo-lters of them is Boolean lter.
References
[1] A. Iorgulescu, Classes of pseudo-BCK algebras Part I , J. Mult-valued Logic and Soft
Comput. 12, 71-130 (2006).
[2] A. Iorgulescu, On pseudo-BCK algebras and porims , Sci. Math. Jpn. 16, 293-305 (2004).
[3] A. Iorgulescu, Pseudo-Is eki algebras. Conectons with pseudo-BL algebras , J. Mult-Valued
Logic and Soft Comput. 3-4, 263-308 (2005).
[4] A. Iorgulescu, Clases of Pseudo-BCK algebra: Part I , J. Multiple Valued Logic and Soft
Comput 12 (2006) 71-130.
[5] L.C.Ciungu, States on pseudo-BCK algebras , Math. Reports 10(60), 1 (2008), 17-36.
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[6] G. Georgescu, A. Iorgulescu, Pseudo-BCK algebras: an extension of BCK algebras , Proc-
cedings of DMTCS01: combinatorics, computability and logic, Springer, London (2001)
97-114.
[7] H. K. Abdullah, K. T. Radhy, Some new types of lters in BCK-algebra .
[8] K. Is eki and S. Imai, An introduction to the theory of BCK algebras , Math Japon 23
(1978), 1-26.
[9] W.Wang, H. Wan, K.Du, Y. Xu, On open problems based on fuzzy lters of pseudo
BCK-algebras , Journal of Intelligent and Fuzzy Systems 29 (2015) 2387-2395.
[10] X. H. Zhang, H. J. Gong, Implicative pseudo-BCK algebras and implicative pseudo
Filters of pseudo-BCK algebras , 2010 IEEE International Conference on Granular Com-
puting (2010) 615-619.
[11] W. Wei, Solution to Open Problems on Fuzzy Filters in Logical algebras and Secure
Communication Encoding Scheme on Filters , College of Science, Xi'an Shiyou University,
(2018), 1679-1686.
[12] W. Wei, W. Hui, D. Kai, X. Yang, On open problems based on fuzzy lters of pseudo
BCK algebras , Journal of Intelligent and Fuzzy Systems 29 (2015) 2387-2395.
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