Acta Math. Hungar. 63 (4) (1994), 313-322. ON EXPONENTIATION OF n-ARY ALGEBRAS J. SLAPAL (Brno) 0. Introduction For any medial n-ary algebra G and any n-ary algebra H the set of all homomorphisms of H into G again forms an n-ary algebra G H, the so called power of G and H. These powers are studied in the paper. In particular, we give sufficient conditions for the validity of the law (G~) K ~ G HxK and for the composition of homomorphisms to be a homomorphism of G H • K G into K G. We also discover a cartesian closed category of n-ary algebras. In the last section we generalize the results attained on universal algebras. Throughout the article, n denotes a non-negative integer. By an n-ary algebra we understand a pair G = (G,p) where G is a nonvoid set (called the underlying set of G) and p is an n-ary operation on G. For the fundamental concepts concerning n-ary algebras (as special cases of universal algebras) see e.g. [2]. The direct product of a family of n-ary algebras {Gi [ i E I} will be denoted by I-[ Gi. In the case I = {il,i2} we shall write Gil • iEI • Gi2 instead of 17I Gi. Given two n-ary algebras G and H, by Horn(G, H) iEI we denote the set of all homomorphisms of G into H. If G and H are isomorphic, we write G _ H. Let us recall that an n-ary algebra (G,p) is called idempotent if p(x, x,..., x) = x holds for all x C G, and it is called medial if P(p(X11, Z12,.-.,Xln),p(X21,X22,...,X2n),...,p(Xnl,Xn2,...,Xnn)) : = P(p(Xll, X21,... ,Xnl),p(x12, X22,...,Xn2),...,p(Xln,X2n,...,Xnn)) holds for all xij C G, i,j = 1,... ,n. For example, any commutative semi- group is a medial binary algebra. It is evident that if Gi is a medial n-ary algebra for each i E I, then 1-I Gi is also medial. iEI