Internat. J. Math. & Math. Sci. VOL. 19 NO. 4 (1996) 733-736 733 A REPRESENTATION OF BOUNDED COMMUTATIVE BCK-ALGEBRAS H.A.S. ABUJABAL Department of Mathematics, Faculty of Science King Abdul Azlz University, P O Box 31464 Jeddah- 21497, SAUDI ARABIA M. ASLAM Department of Mathematics Quaid-i-Azam University Islamabad, PAKISTAN A.B. THAHEEM Department of Mathematical Sciences King Fahd University of Petroleum and Minerals P O Box 469, Dhahran 31261, SAUDI ARABIA (Received April 26, 1993 and in revised form November 13, 1995) ABSTRACT. In this note, we prove a representation theorem for bounded commutative BCK-algebras KEY WORDS AND PHRASES: Bounded commutative BCK-algebra, ideal, prime ideal, quotient BCK-algebras, spectral space 1991 AMS SUBJECT CLASSIFICATION CODES: Primary 06D99, Secondary 54A 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 proved a representation theorem for a positive implicative BCK-algebra as BCK-algebra of self-mappings which apparently does not possess many algebraic properties. Cornish [2] constructed a bounded implicative BCK-algebra of multipliers corresponding to a bounded implicative BCK-algebra, but no representation of these algebras has been studied there. The purpose of this note is to prove a representation theorem for a bounded commutative BCK-algebra We essentially prove that a bounded commutative BCK-algebra X is isomorphic to the bounded commutative BCK-algebra X of mappings acting on the associated spectral space of X Our approach depends on the theory of quotient BCK- algebras as developed by Is6ki and Tanaka [3] and the theory of prime deals of commutative BCK- 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, where a: _< /if and only if :r /- 0 for all z, /, z E X If X contains an element such that a: _< 1 for all z E X, then X is said to be bounded X is said to be commutative if :r A /-- y A z for all z, / X, where z A / (/ :r) A non-empty set A of a BCK-algebra X is said to be an ideal of X if 0 A and z, :r A imply / A A proper ideal A of a commutative BCK-algebra X is said to be prime if z A 1 E A implies x A or t A It is well-known that every maximal ideal in a commutative BCK-