COMMUTATIVE BCK-ALGEBRAS

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 4 (1996) 733-736

733

A REPRESENTATION

OF BOUNDED

COMMUTATIVE BCK-ALGEBRAS

H.A.S. ABUJABAL

DepartmentofMathematics,Facultyof Science KingAbdul AzlzUniversity, PO Box31464

Jeddah- 21497,SAUDI ARABIA

M. ASLAM DepartmentofMathematics

Quaid-i-Azam University Islamabad,PAKISTAN

A.B.THAHEEM

DepartmentofMathematical Sciences King Fahd University of Petroleum andMinerals PO Box469, Dhahran 31261,SAUDI

ARABIA

(ReceivedApril 26, ¹⁹⁹³andinrevisedform November 13,1995)

ABSTRACT. Inthis note, weprovearepresentationtheoremfor boundedcommutativeBCK-algebras KEY WORDS AND PHRASES: Bounded commutative BCK-algebra, ideal, prime ideal, quotient BCK-algebras, spectral space

1991AMS SUBJECTCLASSIFICATION CODES: Primary 06D99,Secondary54A

1. INTRODUCTION

The representation theory of various algebraic structures has been extensively studied The corresponding representation theory for BCK-algebras remains to be developed. Rousseau and Thaheem provedarepresentationtheoremforapositive implicative BCK-algebraasBCK-algebraof self-mappingswhichapparentlydoes notpossessmany algebraic properties. Cornish [2] constructeda bounded implicative BCK-algebraof multipliers correspondingto aboundedimplicative BCK-algebra, but norepresentation of thesealgebrashas beenstudiedthere. The purpose ofthisnoteis to provea representation theorem for^abounded commutativeBCK-algebra Weessentiallyprovethatabounded commutativeBCK-algebraX isisomorphictothe bounded commutativeBCK-algebraXof mappings acting onthe associatedspectral space ofX Ourapproach depends onthe theory of quotient BCK- algebras ^asdeveloped by Is6ki and Tanaka [3] and thetheory ofprime deals of commutativeBCK- algebras Before we develop our results, we recall some technical preliminaries for the sake of completeness A BCK-algebra is a system

(X,

,,0,

_<) (denoted

simply by X), satisfying (i)

(z,y),(z,z)_<z,y

(ii):r,(z,y)_</ (iii):r_<z (iv) ^0_<:r

(v) a:_<y,y_<z

implyz-y, wherea:

_<

/if^andonlyif:r /-0forallz,/,zEX IfXcontains anelement such thata:

_<

1for all z E

X,

thenXissaidtobe bounded Xis saidtobecommutative if:rA/--y A zforallz,/

X,

wherezA /

(/ :r)

Anon-emptysetAofaBCK-algebra Xis saidtobe an idealofXif 0 A and z, :r

A

imply/

A

A properideal

A

ofacommutativeBCK-algebra Xissaid tobe primeif zA 1 E

A

impliesx

A

ort

A

Itiswell-known thateverymaximal idealin a commutativeBCK-

73/ tI A SABUJABA!,.M SAd.AM^AN])A B TIIAHEF.M

algebra is prime (see eg [4]) The theory ofprime ideals plays an important role n the study of commutative BCK-algebras For some information aboutprime ideals, we referto [5] which contains furtherreferences aboutthetheoryofprimeideals AsubsetSofacommutativeBCK-algebrais said to be A-closedifx A y ESwhenever x, yES

Wenowstatethefollowingtheorem known as theprimeidealtheorem(see [6, Theorem 24] and [5,Corollary³])

THEOREMA. Let IbeanidealandSbea A-closedset

of

^acommutativeBCK-algebraXsuch thatSN

I .

^{7henthereexistsaprimeidealPsuchthat}

^I

^{C_ PandP S}

.

COROLLARY B. Let

I

be an ideal

of

^a commutativeBCK-algebra

X

^{and a}

X

such that a I. Then thereexistsaprimedeal

P

suchthata

P

andI

c_

P.

Theabove corollary follows from TheoremAby choosing^s

{a}

^Ifa non-trivial commutative BCK-algebra and I

{0},

then Corollary B ensuresthe existence of a prime ideal in X We now recall the definition of a quotient BCK-algebra IfX is a BCK-algebra and A is an ideal ofX, then we define an equivalence relation ^,-on X by xy if and only if x,y,

y,xA

Let

C={yX’x.y,y.xA}

Let

C’={yEX’x.,y.xA}

denote the equivalence class containing^z X Thenone can seethat

Co ^A

^and

C Cy

ifand onlyif x y Let

X/A

^denote

theset of all equivalence classes

C,

x EX. Then

X/A

^{is a}BCK-algebra(known asquotientBCK-

algebra)

^with

^C C ^C,,

^and

^{C <} C

^{ifand onlyif x y}

^A,

^and

^{Co A}

^isthezeroof

^X/A

(seeforinstance [3-7]). IfXisbounded commutative, then

X/A

^isalsoboundedcommutative with

C

as the unit element For the general theory ofBCK-algebras and other undefined terminology and notations usedhere,werefertoIs6kiandTanaka[3-7]and Cornish[8]

2. A REPRESENTATION THEOREM

Throughout

X

denotes a bounded commutative BCK-algebra. Let

Spec(X)

^{denote the set of} all prime ideals of

X,

called the spectrum of

X

It has been shown in

[5]

^that Spec(X) is a compact topological spacereferredto as the spectral space associated with X. It iswell-known that

f"l

^P

^{0}

^{(seee.g [81).}

PSpec(X)

DEFINITION2.1. For

an

^{x E}

^X,

^{wedefine amapping}

.s(x) [.j x/P

PSpec(X)

where

(P)

^{denotestheimage ofxinto}

X/P

Itiseasytoseethat

(P) Co

^{ifand onlyif x P.}

We denoteby

X,

the set of all mappings

,

x EX. Forany

,

yE

X,

wedefine the following operationsonX

(x

y) and

_<

ifandonlyif

.

These operationsarewell-definedbecauseof the properties of quotientalgebras. Indeed,as

(P)

^is

the canonical image ofx in

X/P,

namelythe class

C

relative to

P,

andthe union

X/P

^is

PSpec(X)

disjoint

Routine verifications similar to onesfor quotientBCK-algebras(seeeg [3])leadtothe following PROPOSmO

.z. (2, ,,)

,o,,,,da omm,,t,aveBC-geb,.

Wenowprovethefollowingrepresentation result.

THEOREM2.3. The_mapping x

X

^---, Xtsanisomorphtsm.

PROOF. That is surjective homomorphism follows from the definition(because the mapping x X

C X/P

^{is thecanonical}homomorphism) Toprove that ^isinjectiveit isenoughtoshow

AREPRESENTATIONOFBOUNDED COMMIJTATIVE BCK-ALGEBRAS 735

that

(x)

^ifand onlyif x 0 For any PESpec(X),

(x)(P)

implies that x E P for all P Spec(X)and hence x

["1

^P

{0}

Thusx 0 Thiscompletestheproof

We provideanexample

t

explainsome essential ideasdevelopedabove

EXAMPI,E Z.4 ([3,p 363]) LetX

{0,

a,b,

1}

be aset Define a binaryoperation onXas inTable

0 a b 1

0 0 0 0

a 0 a 0

b b 0 0

1 b a 0

Table

The

(X, ,, 0)

^{is abounded} commutativeBCK-algebra withP

{0, a}

^{and Q}

{a, b}

^asprime

ideals(cfTable2)

0 a b 1

0 0 0 0

0 a 0 a

0 0 b b

1 a b 1

Table 2

Then Spec(X)=

{P,Q},X/P= {{O, aI,{b, 1}},X/Q= {{O,b},{a, 1}},X/P,X/Q,

are disjoint and

U ^x/P

^isthe disjointunion as definedabove The restof thecalculations caneasily

PSpec(X)

be madetoget the representation ofXin this case.

ACKNOWLEDGMENT. Theauthorsaregratefultotherefereeforhisuseful suggestionsthat ledto animprovement ofthepaper Oneoftheauthors(A B Thaheem)thanksK F U P M forproviding researchfacilities.

REFERENCES

[1] ROUSSEAU, R. and

THAHEEM,

A B, A representation of BCK-algebras as algebras of mappings,Math.Japontca,34(1989),421-427

[2] CORNISH, W

H.,

Amultiplierapproach^toimplicativeBCK-algebras,Math. Sere.

Notes,

^$(1980), 157-169

[3]

ISIKI,

K and

TANAKA,

S.,IdealtheoryofBCK-algebras,Math.

Japomca,

21(1976),^351-366

[4] PALASINNSKI, R, Ideals in BCK-algebras which are lower semilattices, Math.

Japomca,

²⁶ (1981),^245-250

[5] ASLAM,

M and

THAHEEM, A.B.,

On spectral propertiesofBCK-algebras,^Math.

Japomca,

38

(1993),

^1121-1128

[6] ASLAM, M and THAHEEM,

A.B,

Anewproofoftheprimeidealtheorem forBCK-algebras, Math.

Japomca,

³⁸(1993),^969-972

[7]

ISIKI,

K andTANAKA, S., Anintroductiontothe theory of BCK-algebras,Math.

Japomca,

²³ (1978), ^1-26

[8] CORNISH,

W.H.,

Algebraic ^structures and applications, Proc.

of

^theFtrst Western Austrahan

Conference

^onAlgebra,Marcel-Dekker,Inc,NewYork(1982), 101-122