Advances in Pure Mathematics, 2011, 1, 367-377
doi:10.4236/apm.2011.16066 Published Online November 2011 (http://www.SciRP.org/journal/apm)
The Localization of Commutative Bounded BCK-Algebras
Dana Piciu, Dan Dorin Tascau
Faculty of Mathematics and Computer Science, University of Craiova, Craiova, Rom ania
Received August 11, 2011; revised October 20, 2011; accepted October 30, 2011
Abstract
In this paper we develop a theory of localization for bounded commutative BCK-algebras. We try to extend
some results from the case of commutative Hilbert algebras (see [1]) to the case of commutative BCK-alge-
bras.
Keywords: BCK-Algebra, Commutative BCK-Algebra, Algebra of Fractions, Maximal Algebra of
Quotients, ∨-Closed System, Topology, Localization Algebra
1. Introduction
In 1966, Y. Imai and K. Iséki introduced a new notion
called a BCK-algebra (see [2]). This notion is originated
from two different ways. One of the motivations is based
on the set theory (where the set difference operation play
the main role) and another motivation is from classical
and non-classical propositional calculi (see [2]). There
are some systems which contain the only implication
functor among the logical functors. These examples are
the systems of positive implicational calculus, weak
positive implicational calculus by A. Church, and BCI,
BCK-systems by C. A. Meredith (see [3]).
In this paper we develop a theory of localization for
commutative (bounded) BCK-algebras, and then we deal
with generalizations of results which are obtained in the
paper [1] for case of Hilbert algebras. For some informal
explanations of the theory of localization for others
categories of algebras see [4,5].
The paper is organized as follows: in Section 2 we re-
call the basic definitions and put in evidence many rules
of calculus in (commutative) BCK-algebras which we
need in the rest of paper. In Section 3 we introduce the
commutative BCK-algebra of fractions relative to a
∨ -closed system. In Section 4 we develop a theory for
multipliers on a commutative (bounded) BCK-algebra. In
Section 5 we define the notions of BCK-algebras of frac-
tions and maximal BCK-algebra of quotients for a com-
mutative (bounded) BCK-algebra. In the last part of this
section is proved the existence of the maximal BCK-
algebra of quotients (Theorem 29). In Section 6 we de-
velop a theory of localization for commu tative (bounded)
BCK-algebras. So, for commutative (bounded) BCK-
algebra A we define the notion of localization BCK-al-
gebra relative to a topology
F
on A. In Section 7 we
describe the localization BCK-algebra
F
A
in some spe-
cial instances.
2. Preliminaries
In this paper the symbols ⇒ and ⇔ are used for logical
implication, respectively logical equivalence.
Definition 1 ([6]) A BCK-algebra is an algebra


,,1A of type (2,0) such that the following axioms
are fulfilled for every ,,
x
yz A:
(a1) ; 1xx
(a2) If 1
x
yy x, then
x
y;
(B)



 


1xyyz xz
;
(C)


x
yzy xz;
(K)


1xyx
.
In [7] it is proved that the system of axioms {a1, a2, B,
C, K} is equivalent with the system {a2, a3, a4, B},
where:
(a3) ; 11x
(a4) 1
x
x.
For examples of BCK-algebras see [6-8]. If A is a
BCK-algebra, then the relation ≤ defined by
x
y

iff
is a partial order on A (which will be called
the natura l order on A; with respect to this order 1 is the
largest element of A. A will be called bounded if A has a
smallest element 0; in this case for
1xy
x
A we denote
*0xx

. If


x
yyy xx
for every
,
x
yA

, then A is called commutative (see [5,9,10]).
We have the following rules of calculus in a BCK-al-
gebra A (see [6,7]):
(c1)
x
yx

;

D. PICIU ET AL. 368
(c2)

x
xy
 
y;
(c3)
x
yy yx

y

;
(c4)
 
x
yzxz  yz xy ;
(c5) If
x
yzAzxz , then for every ,
and
y
y
zx z.
Proposition 1 ([9], p. 5) If A is a commutative BCK-
algebra, then relative to the natural ordering, A is a join-
semilattice, where for ,
x
yA:
 
x
yxy yyx x  .
Lemma 2 Let A be a commutative BCK-algebra. For
every ,,
x
yz A there exists




x
zyz
 
and
(c6)
 
x
yzx
,
zyz .
Proof. Since
x
yxy by (c5) we deduce that

,
x
yz
,tx zy 
xzyz 
z,
. Let now such that
. Then
tA
x
ytyt z 

z x
, that is,

txy

z
 
x
yzxzy z. □
In [9] (Theorem 8) and [8] (Remark 2.1.32) it is
proved the following result:
Theorem 3 If A is a BCK-algebra, then the following
assertions are equivalent:
1) For every ,,
x
yz A,

x
yz xyxz;
2) For every ,
x
yA,

x
xyxy
;
3) For every ,
x
yA,
  
 
.
x
yyxxy
xy y
 

x
A BCK-algebra which verify one of the above equiva-
lent conditions is called Hilbert algebra (or positive im-
plicative BCK-algebra).
If A is a bounded BCK-algebra, we have the following
rules of calculus in A (see [6]):
(c7) If
x
y, then **
y
x
**
;
(c8) **** ,
x
xx
**
x;
(c9)

**
** **
,.
x
yyxxy xy
Remark 1 If A is a bounded commutative BCK-alge-
bra, then for every
x
A,
 
**
000 ,
x
xxx x
that is, A is an involutive BCK-algebra (see [6], p. 115
and [9], p. 10).
For 1,,,
n
x
xx A () we will define 1n
  
112
,,; .
nn
xxxxx xx 


For two elements ,
x
yA and a natural number
we denote
1n


,, ,;
n
x
yxx xy where
indicates the number of occurrences of
n
x
. Clearly, if A
is a Hilbert algebra, then n
x
yxy, for every
.
1n
Let A be a BCK-algebra. A deductive system (or i-filter)
of A is a nonempty subset D of A such that 1D

and
for every ,,
x
yA

if ,,
x
xyD then .
y
D

It is
clear that if D is a deductive system,
x
y and ,
x
D

then .
y
D

We denote by the set of all de-
ductive systems of A. For a nonempty subset

Ds A
,
X
A
we denote by



:
X
DDsA XD (
X
is
called the deductive system generated by X). It is known
that




11
:, ,
X
xA ;1,for some ,,
nn
.x xx
AxxX In particular for a

, we denote by a the deduc-
tive system generated by


a (a is called principal
and


:1x,for some 1ax n
n
Aa ).


Lemma 4 Let A be a bounded BCK-algebra and
,
x
yA

such that there exists
x
y in A. Then there
exists **
x
y

and

*.
**
x
yx
*y
**
,. Proof. Clearly,

x
yxy Let tA

such
that Then
**
,.txy

*
****
,.

x
yt x
**
tt

**
y
 
*
.
t txy  
 
xyt x 
From (c8) we de-
duce that that is,

**
,y
x
yxy□
Definition 2 ([7], p. 944) Let A be a bounded BCK-
algebra. An element
x
A

is called boolean if


*1xx  (clearly, *
x
xA).
We denote by


BA the set of all boolean elements
of A; clearly,


BA0,1.
Lemma 5 ([7]) Let A be a BCK-algebra. Then for
every ,
x
yA

, .

11xyx y
Corollary 6 For a bounded BCK-algebra


x
BA
iff
*1.xx
Remark 2 If


x
BA
*
* *
10,
, that is, then
from Lemma 4 we deduce that
*1,xx

**
*
xx xx


****
xx xx 0xx
hence
*
0,

 that is, *
x
is the
complement of
x
in A.
Boolean elements also satisfy several interesting
properties which can be proved using above corollary
and some arithmetical calculus:
Proposition 7 ([7]) Let A be a bounded BCK-algebra.
Then for every


aBA and ,
x
yA we have:
(c10)


;A
*
aB
(c11)


;xaxaa
(c12)



 
;x ay
***
,;aaaa aa 
ax
ya
(c13)
(c14) ** ;aa

(c15)


ax ;aa
(c16)




;xaaax x
(c17)




**
ax aaa 
**
;x
(c18) If


bBA, then


;ba aab b
(c19)


;axx 
*
axax

*
***
.ax ax (c20)
Corollary 8 ([7]) Let A be a bounded BCK-algebra.
Then

D. PICIU ET AL.369
1) If , then

aBA

 
:;aaxAax 
2) For ,

A
 
,,0,1BA 
,ab B

;abBA

3) is a Boolean algebra (where for
, and .

B Aab,ab * a b

*
*
ab a b 
Corollary 9 Let A be a commutative BCK-algebra.
For every and

aBA,
y
zA

we have:
(c21)

ayz

y

.a az 


Proof. By (c6) we have
    


 

1
ayaza azyaz
yazyazy azz

   
so (c21) is equivalent with (*)



.ayzy az

az

.az
Clearly, and from
So to prove (*) let
such that and
aazy
zyzy
at
za
tA
y
zt. We have the
intention to prove that
 


 
** .
yazty azz
az yzt


t
Indeed, from





 

 
15
1.
B
c
yztazyz
az ttaaza
ta aat ttt
 
 



□
Proposition 10 Let A be a commutative BCK-algebra.
Then for every and

,abB A
x
A

.b x
we have:
(c22)
 
ax a

bx
Proof. By (c6) we have


  
1
ax bx
abx xbx
abxa xbb

  

  
 .
Also
   

  

 
12
.
C
c
abxx abab
axb ab
axbb

 
 
□
Definition 3 If 12 ,
A
A are BCK-algebras, then
12
:
f
AA

is called morphism of BCK-algebras if

,

f
xyfx fy for every 1 ,
x
yA

(if
12
,
A
A

f
are bounded BCK-algebras, then we add the con-
dition ). 00
3. Commutative BCK-Algebra of Fractions
Relative to a ∨-Closed System
In this section by A we denote a commutative bounded
BCK-algebra.
Definition 4 A nonempty subset of A will be
called ∨-closed system of A if and
S
S0
x
yS
for every ,.
x
yS

For a ∨-closed system we define the binary
relation
SA
S

on A by


S
,xy

 iff there is


s
SBA such that .
s
xs y
Proposition 11 The relation S

is a congruence on
A.
Proof. Clearly S

is an equivalence relation on A. To
prove the compatibility of S

with the operation →, let
,,
x
yz A

such that


S
,xy

 (hence there is


s
SB A such that
s
xs y). By (c21) we
deduce



 

,
szx szsx

s
zsyszy

 
and similarly,


,

s
xzsyz that is,


S
y,zxz

 and


xzyz,.
S

 □
We denote


S
AS A

; the commutative BCK- al-
gebra


A
S will be called BCK-algebra of fractions of
A relative to S. For
x
A

we denote by


S
x

the
equivalence class of
x
relative to .
S

Clearly, in


A
S, 1 =






,11:xAx
SS


 = {
x
A: there is


s
SBA 1 such that
s
x},
0 =






0:xA ,0x
SS


 = {
x
A: there is


s
SBA such that
s
xs} = {
x
A: there is


s
SBA such that
x
s} and for every ,
x
yA

,






SS
x
yxS y


 .
Proposition 12


A
S is a bounded commutative BCK-
algebra, when 0 =


S
s

with

s
SBA

.
Proof. Clearly, if ,
s
tSBA, since


ArstSB and




.rtst

 
S

To prove that, for S
rs

s
SBA

,

S
s

= 0, let .
x
A

We have














SS SSSSS
s
xsxxsxx

 

which is true since


s
sx sx 

. □
We denote by

:pA AS
S the canonical surjec-
tive morphism of BCK-algebras (defined by




,
S
px xS

for every
x
A).
Remark 3 Since for every


s
SBA


, }
we deduce that
0sss



0.
S
pS BA
Proposition 13 If
x
A

, then






S
x
BAS

 iff
there exists


s
SB A
such that So,
if
*1.xxs


,
x
BA then





.
S
x
BAS

Proof. For

x
A

, we have
    

*
*
11
SS
S
xBAS xx
xx




 

S

there exists


s
SBA
11. s
such that
If
*
xx s

,
x
BA since

D. PICIU ET AL. 370
*01xx and


0SBA , we deduce that





S

.
x
BA



S □

A
S
:
verify the following property of universality:
Theorem 14 For every bounded commutative BCK-
algebra B and every morphism of bounded BCK-alge-
bras
f
A

B such that there
exists a unique morphism of bounded BCK-algebras
  
0,BAfS

:
f
AS B
 such that S
f
pf
.
Proof. Let ,
x
yA such that




.xy
S

S

Then
there is


s
SBA


 
00
sy fsx
fx fs
 such that

 
 
x fsy
fs fy
.
s
f
xf
 

 yfxf
 

y

So,

:
f
AS
B defined for
x
A by
S



f
x
fx

is correct defined. Clearly,
f

is
morphism of bounded BCK-algebras and S
f
pf


.
The unicity of
f
 follows from the fact that is
onto. □
S
p
Example 1 If A is a bounded commutative BCK-al-
gebra and


0S S or is such that 0S

and
, then for


\0SBA


 ,
x
yA,

,xy S


00 ,
x
yxy hence


.
A
SA
Example 2 If A is a bounded commutative BCK-al-
gebra and is an ∨-closed system system such that
(for example or ), then for
every
S
1S
,
SA

SBA
x
yA,

,S xy

 (since 11
x
y

and
hence in this case

1SBA,



1AS .
Definition 5


A
S
TA
is called the BCK-algebra of
fractions of A relative to S.
4. Multipliers on a Commutative Bounded
BCK-Algebra
The concept of maximal lattice of quotients for a dis-
tributive lattice was defined by J. Schmid in [11,12]
(taking as a guide-line the construction of complete r ing of
quotients by partial morphisms introduced by G. Findlay
and J. Lambek (see [13], p. 36). The central role in the
construction of the maximal lattice of quotients for a
distributive lattice due to J. Schmidt in [11] and [12] is
played by the concept of multiplier for a distributive lat-
tice defined by W. H. Cornish in [14].
In this section we develop a theory for multipliers on a
commutative bounded BCK-algebra A.
Definition 6 A subset is called ∨-subset of A
if for every and
aA
x
T

we have . axT
We denote by

TA

T A the set of all ∨-subsets of A.
Clearly (and more generally, if denote
by

Ds A


I
A
 
the set of all increasing subsets of A, then

I
ATA
Remark 4 ). Clearly, if 12
DD

,,TA then


12 .DTA D
Lemma 15 If


,TA then D
1) 1;D

x
D

2) If
x
y

and , then .
y
D
Proof f . (i). I
x
D

, si,nce 1
A
then 11 .
x
D
3) We have .yxy □


partial strong multiplier on A we mean Definition 7 By
a map :,
f
DA where

,DTA such that:
(sm1) For every
x
D

a

,A nd

eB


f
e fx; x e
(sm2) For every
x
D

,

x
fx;
(sm3) If


BA,eD then
 
;
f
eBA
(sm4) For every
x
D

and e

,DBA


.

f
e fx xe
By




A
f
we denote the domain of dom fT; if


do fA

, we called
f
total.
the languag we wil
m
To simplify
ins
ps 0
e,l use strong multiplier
,1
tead partial strong multiplier using total to indicate
that the domain of a certain multiplier is A.
Examples
defined by 0

x
x

:
A
A1) The ma and
respectively 1


1x

, for every
x
A are total strong
multipliers on A.
2) For


aBA and the map

,DTA
:
a
f
D by A defined


a
f
xax, for every
x
D

is a strong multiplier on Aipal).
If
(called princ


,
a
dom fA

we denote a
f
by a
f
.
Re :mark 5 If
f
DA is a strong multiplier on A
(with


DTA), then


1. Indeed, if in (sm1) we
put e1f 1

, we obtain that for every
x
D,






11 11.fx fxf  
For


,DTA we denote




:
M
, :DAf is aAfD
and
strongmultiplieronA




() ,. ADT
M
AMD A



,
ii
f
MDA


A
12
,DD T, 1,2,i

For and we
de 12 : fine 21
f
fD DA by 






21 , 2 1
f
fx fxfxevery  for 12 .
x
DD



Lemma 16 1212 ,.fMDDA
f
Proof. If 12
x
DD and

eBA, then










 




 
21
2
1 2
121
12 2
1 2
,
,
c
sm
ex fexfex
efxe fxfx
fx
fxfxfx x
fefeBA
 
 



 



 
12
12
12
12
f f
efx
ef
ffx
ffe

 


by Corollary 8 (since
2


12
,

f
efe BA and if

D. PICIU ET AL. 371


 

   

   

 


21
4
22
12 1 2
12
12
12 12
,
,
c
sm
c
BA
ef fxefx fx
efx efx
xfe xfe
since if ,
x
yA

such that for every exey




,eA ABA B then in particular, for 0e

we
obtain 00.
x
yxy
12
eD D
x
fefexffe

 
 
 

If A is bounded,

DTA and then is
regular. We denote
0D,D




:.RADAD isaregular subsetofA
Lemma 19 If




12
,,DD TARA then




.ARA
12
DDT


Proof. By Remark 4, 12 Let
.ADDT,
x
yA

such that for every exey


.BA

12 12
,.
f
fMDDA 
llary 17
 
,,0,1MA
that is, □
ounded commu-
at
Coro is a b
tative BCK-algebra.
12
For every
eDD


,BA 1, 2,i
ii
eD since


12
ee D
1 2
D BA  we have

M
A
ma 16
Proof. The fact this a commutative BCK-
algebra follows from Lem. If

,DTA

,
f
MDA and ,
x
D then

101
x
xf
order on
x and sxince the relation of

M
A is given by 12
f
f iff



12
f
xfx
for every



,
12
x
do
that is,
m fdome ded
01,ff

 wuce that
M
A is bou
p
nded. □
18 TheLemma ma


:
ABAMA defined vby





12121 2
1222,
eexeeyeex
eey exeyxy




so


12 .DDRA □
We denote








:.
r
M
AfMAdomfTAR A 

Aa
a f for every v


aBA is a morphism of
K-algebras.
Proof. If

,abB A and
Corollary 20


r
M
A is a BCK-subalgebra of


M
A.
bounded BC
,
x
A then


Proposition 21

  




r
M
A is a Boolean subalgebra of


M
A.

22
,
a b
c
ab
fxf fx
axbxabx fx

  
so,
ab
f x
Proof. Let :
f
DA be a strong multiplier on A with




RADTA. Then*:
f
DA,








0,
*
f
xfxfxx for
x
D.
 
AAA
vavb vab
and
We have

0
00
A
vf .□
finition 8 called regular if for every De is








   
*
ff xfxfxx
f
x xfxfx




DA
,
x
yA such thaey for every

,BA then
t e x
.eD
x
y
nded BC

ple, a bouFor examK-algebra AThen for


eDBA and
x
D we have
is regular
    

  

   










   
21
4
22
c
sm
c
ef fxefxxfxfx
efx exefxefx
xfeexxfexfe
xfe efefexfe

 

 



 




 




 


 
 
 
*
1111.
efe
xfeefexe ex
*






 


Since we deduce that

DRA
 
1
if



21
dom fdom fextends 2
f
1
f
we say that and




2
,
1
f
xfx for all

2;
x
dom f we w21 rite
f
f

if 1
f
extend2 s
f
. A srong mtultiplier
f
is called
maximal if
f
can not to a strictly larger
don.
Lemma 22 1) If

12
,
be extended
mai
f
fMA,


r
f
MA and

*,
f
f hence * ffx x1, that is,


r
M
A
is a Boolean algebra (by Coro
Remark 6 The axioms sm3, sm4 were necessary
pr
llary 6). □
in the
oof of Proposition 21.
Definition 9 Given two strong multipliers 12 ,
f
f on A,

D. PICIU ET AL. 372
1
f
f, 2
f
f, then 1
f
and 2
f
coincide on the
 
12
;mfdom f
veg multiplier

r
do
2) Ery stron
f
M
ltiplier. M
A can be ex-
tended to
ea
a maximal stronmre precisely, g uo
ch principal strong multiplier a
f
with


BA and
  
a
domfTAR A can be uniquely extended to
the total strong multiplier
a
a
f
each cipal
extended to a maximal non-
principal one.
Proof. 1) If by contrary, there exists

1
tdomf
and non-prin
strong m can

om
ultiplier
d
be
suc

f
2
h that
 
12
,
f
tft

since
 
mfR A, then there exists do


BA td
omf
such that





12
ft tt
 

ctory, since tt

2) We fira
12
tt f



.domf
tf
ontradi
st pr
t

ove
t f
that
which is c
f
with

aBA can 
mu
not be
extended to a non-principal strong ltiplier. Let
  
a
domfTAR, :
a

D A
f
DA
l stro
and suppose
by contrary that there exists

,DTADD

, (
-png multiplier

,
hence

A R

A) andDT
 a nonrincipa
f
MDA

 which extends a
f
. Since
f
is non-
ts 00
, principal, there exis
x
Dx D

 such that
00
.

f
xax Since


,DRAthen there exists

tDBA such th
 
0
,axx atx  t
a
at

x t
is cont

sin

000
ftf
which

radictoryce .
f
f Hence a
f
is
uniquely extended by a
f
.
Now, let

r
f
MA be non-princi and
 
:
pal
 
,, ,,
f
 
M
DgM DAdom fgDT AD 
and


fdom
g
f (clearly, if


,,
f
Dg M then




D A).
Th
T
e s
A R
et
f
M
is ordered by




1122
,,DgDg iff
12
DD and 11
2D.
g
g Let


,:
kkk K be

a Dg
chain in .
f
M
Th

en 'k
kK
DD

 
.D  So,

TA and

dom f:
g
DA


 
defined by

k
g
xg
 if
xk
x
D is correctly d(sinefined ce if
kt
x
DD with k by 1),

,tK, then


kt
g
x

gx ).
Clearly,

,
g
MDA and

 dom f
g
f
 (since if
,

x
dom fD then
x
D  and ts so there exis
kK, suchk that
x
D, hence
 


k
g
xg

xfx).
n upperfor the family

:
k
gk Kce by Zo
So,

,Dg
 is

a bound

D,
k, henrn’s lemma,
f
M
ex
co
h
n-
tains at least one maximal strong multiplier h whic
.f is non-principal and htends
f, h is also non-principal. □
On the Boolean algebra

r
extends e .f Sinc
M
A we consider the re-
A
lation

defined by

12
,ff A

 iff and co-
in
1 2
cide on the intersection ofmains.
Lemma 23 A
f f
their do

is a on

r
couencengr
M
A.
Proof. The reflexivity and the symmetry of A

are
im f let




12 23
,,,A ff ff

. Therefore 12 ,ff, and respevely
23
,ff coincide on the intersection of theomain
 
01 3
cti
ir ds. If by
contrary, there exists

x
dom fm f such that do




10 30
,
f
xfx since
 
2,domfR A there exists




2
tdomfB A such that



 
1 0
x ftxx

0

3
ft
since
10 30
tfx tf
which is contradictory,





.m f
mediately; torove the transitivity o p A

01 2
bility of A
3
f do The txdom fdom compati-

with → on


r
M
A is immediately. □
For


r
f
MA we denote by


f
the congruence
class of modulo A

and


f A
 .
rA
MA


Rem Proposition 21 weduce that ark 7 Frome d
A

 is
a Booleagebra. n al
Lemma 24 The map

:
A
vBAA

 defined by

Aa
vaf




 injecti of Boolean al
nd
is anve morphism-
gebras a






.BA RA


fact that
A
v
he
Proof. TA
ma 18. ove th
v is a morphism of Boolean al-
gebras folloTws frome injectivity Lemo pr
of A
v let


,abB A ch that su



AA
va vb. Then


 
,
ab Aa
ff fffxfx 


 , or every
abb
,
x
A,
x
Axaxb

 for every hence for
0x

we obtain that 00ab .ba To prove






,
A
vBARA


 if by contrary tht ere exis


r12
,
f
fMA such that




12
f
f t (that ishere exists




201
x
dom fdomf such that



10 20
f
x

fx)
and

12 12aaaa
ffffffff

 
 




for every




12
,aBA fxaxfxax

 for
every


aBA and every



.
12
x
dom fdomf
For 0a

and 0
x
x

we obtain that




100 20010

20
f
xxx f
contrry. □
xfx fx which
to
is
adic

aBA, a
f
Remark 8 Since for every i
n
s the
unique maximal str a f


ong multiplier o (by Lemma
22) we can identify a f



 with a
f
. Sce o, sinA v is in-
jective morphism of lean algras, the elemnts of Booebe


BA can be identiwith th elements of the set fied e



:.BA
a
fa
Lemma 25 In of the identifications made above,
if
view


f
A


 (with


r
f
MA and






domfTA RA), then
 
D

 

.DBAaBAf BA
Proof. Let
:
a
f


.aDBA If by contrary,




BA then
a
ff a
f
f is a non-principal strong
men byultiplier. Th22, (2), Lemma a
f
f c
aximal strong multiplier
an be ex-
tended to a non-principal m

D. PICIU ET AL.373
:
f
DA with

.DTA Thus, DD and for
every ,

x
D


 
.
a
f fxfxx


x fxaaf 
Since , then

aDBA
 
fxxfa
 
,fax  that is,

4
sm
D
f is principal
which is contradictory with the assum tion thatp
f
is
non-principal. □
mmutative BCK-Algebra of
Quotients
Th section is to define (taking as a guide-
ne the case of distributive lattices) the notions of BCK-
5. Maximal Co
e goal of this
li
algebra of fractions and maximal BCK-algebra of quo-
tients for a commutative bounded BCK-algebra. For
some informal explanations of notions of fraction see [13]
and [5].
Definition 10 A bounded commutative BCK-algebra
A
 is called BCK-algebra of fractions of A if:
(f1)


BA is a BCK-subalgebra of
A
;
(f2) For every ,, ,abc Aa b

, there exists
suc

h that aa ab

 an aBA d


.c BA

 a
As a notational co nvenience, we write
A
A

 to in-
t dicate tha
A
 is a Bf fractiCK-algebra oons of
A
. So,
 
A BA (since for

,,abc BA

 with abB



,
if consider

,BA then 00aa b0b


 and

A).
Definition

is the tive)
quo

0cc

 B
11

QA maximal (commuta
BCK-algebra of ts of A iftien


A
QA and for
every commutative anbounded BCK-algebra d
A

with
A
A , there exists a monomorphism-algebras of BCK


:.iAQA 
osition 26 Let A be a commutative and bounded
uch t
Prop
BCK-algebra shat
A
A

. Then
A
 is a Boolean
algebra.
Proof. If by contrary,
A

6, there exists
i o
Corollary
s not a Bolean algebra,
then by
x
A  such that
*1.x Since x
A
A , then there exists


eBA,
such that

exBA and

11.e e 
emma 4,
 
ex exe 
*
xx
Then, by L




***
**
**
11
1
11 1,
xex
exeexx
exx exx

 

a contradiction! □
Remark 9 If A is a Boolean algebra, then


.BA A

By Proposition 26,


QA is a Boolean alge
ax
bra and the
ioms sm1-sm4 are equivalent with sm1, hence


QA is
in this case just the cal Dedekind-MacNeille com-
pletion of A (see [12], p. 687). In contrast to thral
situation, the Dedekind-MacNeille completion of a Boo-
lean algebra is again distributive and, in fact, is a Boolean
algebra (see [15], p. 239).
Lemma 27 Let
classi
e gene
A
A

; then for every ,,ab A



ab



, and any finite sequence 1n
ists
,,cc

,A

 there
ex


aBA sucaa ab

 andh that


iBA for 1, 2,,inac



 (n
Proo lemma hol1. So we
2).
for nf. Assumeds true may


bBA such thatfind ba bb

and


i
c BA

 for 1, 2,,1.inb

 Since
A
A

, we
find


cBA such that

cb

cba



and b



nBA


ed proper
cc . The

cB the
requirties. □
Let
element abA has
.aA

 Lemma 28
A
A

 and Then








TBA   :
aa BABAAR
 .
Proof. If
Daa



aBA, then


x
aBA

 and a
x
D 
and since






ax aaxa

 B A
it follows
,
a
xDa

 he

.
 To provence a

BADT


a
DRA
 consider ,
x
yA

such that exey,
for every


.A
a
D

 ntrary, eB If by co
x
y

, since
A
A

, the re exists a such that

BA
0


0
aaB

 0a D

) and 00 ,axay
s contradictory
A (that
.
is,
□
a
which i
Theorem 29
A

 (definection 4)l
(commutative) BCK-algebra o
ed in S is the maxima
f quotients of A.

QA
Proof. The fact that


BA is a BCK-subalgebra
(Boolean subalgebra) of


QA follows fromemma 2 L4
and Remark 8. To prove

,

A
QA let








,,
f
ghQA wit

,r h ,
f
gh MA such that




g
h (that is, there

exists


0
x
dom gdomh
such that




00
g
x
Put
hx ).




domfTA R

and D A

  

[] :.BA fBA fa
Then
D af


.
f
DBAD by Lemma 25, If suppose
that for every




,
aa
fgfh


,BA then aD
,
aa
fg fh


 


 ev 
hence forery




x
domgdom h we have






logou
aa
fgx fhx (anasly than as in the
proof of Lemma 24)




.xagx ahx  axgx axh 
Since


DRA we



g
xhxdeduce that for
every




x
domgdom h so




,
g
h which is
contradce, if ictory. Hen




,
g
h then there exists


,BAaD such that




.
a
 But for
this
a
fg hf


A we have



a
ffBA aDB(since


[]
f
DBAD

.
A
QA
of QA
) hence
To aximalityt prove the m, le

A

be a

D. PICIU ET AL. 374
bounded commuK-algebrah that
A
A

(t2) If 12 ,DDF

, then
Exam 12.DDF ple 3 If

tative BC suc,
thus
 
;BA BA

 Then
A

isdded i

, then the set D emben


QA
by

:iA QA
 defined by
 

,
a
ia f
 fo
aA



aa
f D

 (see Lemma 28).
Clea
r every
,

where dom
rly,

ar
f
M
obras
of i, let a
A
 (by Lemm is a mor-
f BCK-algeemma 18). To prov
injectivity ,,bA

 such that

a 28) and i
phism (see Le the



 
ab ab
f
ffxfx
 
  for every .
ab
x
DD



If ,ab

 by Lemmce a 27 (sin
A
A )

b BA

 a
a b

 which is contradictoe

b BA

 iab

). □
1. If A is a BCK-algebra with
nd
, th
nd
ry (sinc
aDD


ere exists

A such that
a a

,a


Remark 10
 
2
0,1 L a
aB
aa
BA
,aa
mplies
a

A
A



0,1 , then A hence
TA




D:
F
D DTAD


 
recall tha
is a topological system on A.
denote the set
of all regular subsets of
Example 4 Wet by

RA we
A (see Definition 8). Then




F
TA RA is a topologystem on A (see
Lemma 19). SA a ∨-closed subset of A (see
Definition 4). I
ical s
Example 5 Let
f we denote by




:,
SDTADBAFS

  then S 
F
is a
topological system on A.
is a topological system on A, l
F
If et us consider the
relation
F

of A defined by: there exists

,F xy  
DF

such that txty for any


.tDBA
As in the case of S

(see Prop deduce that osition 11), we
F

is a congruence
We shall denoty
on A.
be

2.A L Indeed, if ,,abc A
 Q

, with a

A sat e(hence

.ecBA 0,e he

,cBA th

.
b, then
b,
nce

there exists eB
) and
at is
uch th
Clearly,
ae
1e
A
F
x

the congruence class of an
ment
BA Asof BCK-
ras wiy we have BCK-alg
hains.
2. More general, if A is a BCK-algebra such that

opert
examples
local algeb
and
th this pr
BCK-c
ebras
ele
x
A

and by :
F
F
pA A

 the canonical
mras.
1 C
orphism of BCK-algeb
Remark 1learly, if



.
FF
BA  
Definition 13 A
aBAa
F
-multip


BA
is finite, if
A
A 
lier on A is a mapping
then

,
A
BA hence
der

d

consi
 
A BA
Indeed, A
 an.aA 

Q

BA


BA be-
ing finite, there exists a smallest element


eBA
, then there

ae
a
a
such a e Suppose a e
would exists

eBA such that

a
ee
and ut ea ,
a
ee

.aBA

.eaBA B
aa

BA implies

and thus we ob

a
eae
a conce

,a that is

.
tain
tradiction. Hen
,ae
,
ea e

aBAa
a
e
A
BA
 Then

,
A
BA
 hence



.BA
ded
QA
Boun
e localization lattice

ation otative6. Localizf
BCK-ALgebras
Com
it
mu
ed th [4], G. Georgescu exhibIn
F
L
of a distributive lattice L with respect to a topology
F
on
see L in a similar way as for rings (see [16]) or monoids (
[17]). The aim of this section is to define the notion of
localization BCK-algebra
F
A
of a commutative bound-
ed BCK-algebra A with respect to a topology
F
on A. In
the last part of this section proved that the maximal
commutative BCK-algebra of quotients (defid in Sec-
tion 5) and the commutative BCK-algebra of fractions
relative to a ∨-closed system (defined in Section 3) are
BCK-algebras of localization.
In this section A will be a bounded commutative
BCK-algebra and
is
ne
F
a topological system on A.
Definition 12 A non-empty family
F
of elements on

TA will be called a topological system on A if the
following properties hold:
If

12
,DFDTA and 12 ,DD then
2
DF (hence
(t1)
A
F);
:
F
fD A

 where DF

such that for every


BA and ,a
x
D

(m1)


;
F
f
a
(m)
x ax   f


.
x
2Ffx
 


, then a
F
F
A-multiplier is a function
ich ve only the conditions sm1 and sm2
If
:Arifyf
fro
A wh
Definitiom n 7. Theaps 0,1: m
F
AA

, defined by
0


F
xx

 and 1


1
F
x

 for every
x
A

are
F-multipliers. Also, for

aBA, :
aF
fD A

 de-
fined by


aF
x aF xf

  for every ,
x
D

is a
F-multiplier (where DF

).
For DF

, we shall denote by

,
F
MDA

t
s havithe domain . If
he set
of all the F-multiplierng D
12
,,DD F

12 DD we have aapping canol mnica




12
,21
:, ,
D
DF F
MDA MDA

  defd by ine


12 1
,DD
D
ff  for

2,. F
MDA   Let us consider f
the directed system of sets


,,MDAF
D
F




F
A
12 121 2
,,, DD DDFD D
 and denote by the inctive
limit (in the category
du
of sets):


li ,
DF
m
F
F
AMDA



.
For any F-multiplier :
F
fD A

y

we shall denote b
,Df the equivalence class of in f
F
A
.
Remark 12 We recall that if :
i
fi F D A

, 1,2i

multipliers, then

F-are 112 2
f(in,,DfD
F
A
) iff
ere exists ,DFD DD th 12

 such that 12 .
D
D
ff

Let :F fD A
ii

, (2), Fulti- with ,1,DFi
ppin
i
pliers. Let usg
-m
consider the ma
1212
:
F
ffDDA

, defined by






12 1 2
,
f
fx fxfx for any 12 ,
x
DD


and let

D. PICIU ET AL.375

11221212
,, ,DfDfD Dff .
This definition is correct. Indeed, let :
ii F
fD A

 ,
with ,1,2
i
DFi

such that

,
ii
Df,
ii
Df

,
Then there such that 1, 2iexist 12 ,DD
 
.
1
D
 
F 
2
D
 and
112 2
,DDD
 
 
D11 ,
DD
 


11
ff
2 222 DD  
12 12
DDD DD
  
.ff
 If we set 
12
DD
 , then DF 

and
clearly

12 12
,
DD
ff ff
 


hence

121 2121
,,DDff DDff2.

 

ma 30Lem

12 12
,.
F
ffMDDA

 
Proof. If 12
x
DD

and n

aBA, the


 





21
1
()
12
/
FF
c
F

2
12
12
f
fax xax f
afxa
affx





and
 

fa
fx


2
12122.
m
F
ffxfxfxfxx

□
rollary 31Co

,,0,1
F
A is a bounded commutative
Bre CK-algebra (whe

0,A0 and

1,1
A
) (see Co-
rollary 17).
Definition 14
F
A
will be called the localization BCK-
algebra of A with respect to the topology
F
.
Lemma 32 The mapping

:
F
F
vBA deAfined by


,
F
a
a Af for every orphism of v is a m
BCK-algebras a

is a re

aBA
nd


F
vBAgular subset of
F
A
.
Proof. If


,ab B A then
 



,
,, .
b
ab abF
v bAf,
FFa
v aAf
A
ff Afvab

 
 
To prove that is a regular subset of

F
vA
F
A
, let

,,
ii F
Df A,1,2i, such that
i
DF




11
,,,
aa22
,
A
fDfAf every
Then
Df for

aBA.

112 2
,,
aa
Df fDff there exists
such that
12
DFDD,D

  
12
12
,
aa
DD
FF
ff
axfx axfx 

 
for every
ff
x
D
oose
and If in this last equiva-

aBA.

BA lence we ch0,athen we obtain that

 
12FF
xf fx
12 1122
,,,
xx
f
xfxDfDf

 
hence
 




F
vBA is a regular subset of
F
A
. □
7. Applications
that follows we describe the localization BCK-algebra In
F
A
in some special in
1) If
stances.


DTA and
F
is the topological system




:
D
F
DTADD


  (see Example 3), then


,
F
AF MDA

 and

:
F
F
A AvB is defined by



,
FaD
v


,
x
yBA aor any

aBA. For Df f
we have


,F xy 

 for every



,x
Dx 
njective m of BC
FF
yD
tDtf v 
there exists an iorphism K-algebras
xty f vy then
:,
F


FFF
A
A


x
vx    such that 
.
F
F
vp



2) To obtain the maximal BCK-algebra of quotients


QA as a locization relative tol system al a topologica
F
we will develop another theory of F-multipliers
(mDef
str
eaning we add new axioms for F-multipliers).
inition 15 Let F be a topological system on A. A
ong-F-multiplier is a mapping :
F
fD A

 (where
DF

) which verifies the axioms m1 and m2 and
(m3) If


,eDBA then
 
F
fe BA

;
(m4)






FF

,
x
fa a   ry fx for eve


BA and .
x
D

aD
If


, then
F
A
F

is the identity congruence of A
so ulti is a strong tot(in
sense of Defini.
mark 13
a str al multiplier
tion 7)
ong F-m plier
Re If A is a BCK-algebra, the maps 0,1:
F
AA

 defined by 0


F
xx

 and 1


1
F
x


for every
x
A

are strong F-multipliers. If
:
ii F
D Af

, (with ,1,2
i
DFi

) are strong F-mul-
mapping tipliers, the 1 221 :
F
f AfDD


by
defined






12 1 2
,
f
f
12
x fxfx for any
x
DD

 is also a stier.
Remark 14 Analog tipliers if
we we obtain
of
rong-F-multipl
ous as in the case of F-mul
work with strong-F-multipliers a BCK-
subalgebra
F
A
denoted by
F
s
A which will be
called the strong localization BCK-algebra of A with
respect to the topological system F.
If




,
F
TA A then R is th
F

e identity con-
gruence of A and we obtain the definition for strong
multipliers on A, so

lim
F
A
M,DA
DF

tion it is easy to see that
. In this situa-
F
v is injective, so we have:
Proposition 33 In t

, he case

F
IARA
F
s
A

is exactly theve BCK-alge- maximal commutati
bra of quotients


QA o (see Section 5, Theorem 2f A9).
call (see
Propos
3. Let S be a ∨-closed sy
ition 11) that on A we have the congruence S
stem of A. We re

defined by:

iff there is


s
SBA such

y,S x


that
s
xsy and


S
AS A

 is called the

D. PICIU ET AL. 376
system S ( from Section
ase we have the topological system
(commutative) BCK-algebra of fractions of A relative to
the ∨-closed see Remark 5 3). In
this c S
F
associated
with ,S
 
:.
S
FDTADSBA 
Lemma 34 .
S
F
S

 
Proof. For ,,
x
yA if

,S
F
xy

 then there exists
DFS such that
s
xsy for every


s
SBA.
ts


DSBA , so this

Si
0
nce S
DF, ere ex
s
DSB partitain
00
,
A; in cular we ob
s
xs y hence ,
S


,xy

 that is, S.
F
S

 
If

S
,xy

, then 00 ,
s
xs 

y for some
0
s
SBA. If :A
pal e system gy

rinci
consider


0
Ds a
deductivenerated

0
sa
0,
(the pb
s
see
Corollary then S
DF

0
8, 1)), (since
s
DSB

A ). If
s
S
00
BA then
s
ssss hence
 

00

,.
S


00
SS
F
SFF S
ss xsx
ssy ys
xy
sx s
ssy

 
 
 

□
tion 35 If S
 
 
 
Proposi
F
is the topological
a ∨-closed subset S of n
sy
A, the
stem on A
associated with S
F
s
A
is isomorphic with



.BAS
Proof. Following Lemma 34, ,
S
F
S

  therefore a
S
F
-multiplier can be considered inis case as a mg thappin


:
f
DAS (DS F) having for
x
D and


DBA the properties a


 
S

 
,,
.
S
S
fa xafx
x
fa



 x
fa BAS


If
fx  
 

112 2
,, ,lim,
SDF
S
F
DfDfs A,AS


 
and
MD

112 2
,,Df Df then there exists such
th and
S
DF
at 12 DDD 12 .
D
D
f
f

Since
n
12
,,DS F, the

DD



12
,SBA DA 
are nonem, hence th
,SBA DBD S
ptyere exist


,
s
D

SBA
11
s
DSBA and

22
s
DS

BA.

. We shall prove that 112 2
f
sfsed, if con- Inde
sider

12 ,tsss DSBA  then
 
1
 
12111SS
f
tss   fs fs (since
2SS
ss

 0) and an

alogously
 


22112
.
2 2
f
tn a simifsfs fs
 
112 2
Ilar way we
can show that
f
tft for a
,.ttSB
ny
with tese consid-
mapping

A In accordance
12
erations we can define the
Dh


  


:l

im,
SDF
S
F
s
AM BA   DAS

 S
by putting

,,Dff s   where

.
s
DSBA It is easy
is a mebras. We
shall prove that
orphism of BC

is injective and surjectiv
the injectivity of
e. To prove

112 2
,, , let

S
F
DfDfs A such
that




112 2
,,.DfDf   Then for any


,
to prove that

K-alg
11
s
DSB

A22
s
DSBA we have




112 2
.
f
sfs For two fixed elements 12 ,
s
s with


,
ii
s
DSBA 1,2i

, we consider the element


2

.
12 1
s
ss A We have DD SB




0

12111111S
f
s sfs   fs fs and




222222
0,
f
2 1S
s s   henfsfs fs ce




12
.
f
sfs Now t le




ss
12 12
:.DsDD
s
DsD


 Sin   ce
s
s
D

we deduce that .
s
D If aA and
s
s
D


then

.
ss
s
sa s asDD
 
SA Sinc
.
e
s
sS
s
DDDF

  If
s
s
D
, then



11
11
0
S
ss sfsfss
1
s
fsfs fs 

 


 
and analogously,





22 12
112 2 12 ,, ss DD
fs fsfs fs
,
f
fDfDf


 

that is,

is injective. To prove the surjectivity of

, let




SAS
  withaB aA

.
ent
s
S

For fixed elem one we consider ,




:.DsxAsx  Clearly .
S
DF We define


:
a
f
DAS by putting
 
,
aS
fx ax

 for
every .
x
D

Ca
f
ilearly, s a strong S
F
-multiplier
en (clearly (m3) is verified si

,eDBA thnce if







.
SSSa
f
eae a  e    From BAS  


 



,,
a
Dfa
SS
aS S
sas as
fs a
asa




 



that is,


is surjective, hence bijective. □
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