Examples of finite-dimensional Hopf algebras over a field, whose antipodes have arbitrary even orders >4 as mappings, are furnished. The + 1 0 1 0 Y. We define an algebra homomorphism e from R to K by specifying e(X,) = 1 for 1 < i < n, and e(Y) = 0. One easily checks that (e 0) IR)A = IR = (IX 0 e)A (identifying K 0) R and R 0 K with R), as these are algebra homomorphisms agreeing on the Xi and Y, so that e is a counit. Hence, R is a bialgebra. Let I be the ideal of R generated by the union of the following four sets of elements of R:(1) {X1 -11 < i < n} { Ye}We assert that I is a bi-ideal of R, i.e., AI C I 0 R + R 0 1 and e(I) = 0. As A and e are algebra homomorphisms, and since J = I 0 R + R 0 I is an ideal of R 0 R, it suffices to check these conditions on the generators of I. Since e(X,) = 1, we have e(Xi" -1) = e(XiXj -XjX1) = 0, and since e(Y) = 0, we have e(YX1 -wXiY) = e(Yq) = 0. Working modulo J, A(Xiq-1) = Xiq Xiq-1 1-1-01 = 0. For (2),

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