We show that there exist sequences ([L.sub.n] | n [TEXT NOT REPRODUCIBLE IN ASCII] k) of right, left, and two-sided regular ideals, where [L.sub.n] has quotient complexity (state complexity) n, such that [L.sub.n] is most complex in its class under the following measures of complexity: the size of the syntactic semigroup, the quotient complexities of the left quotients of [L.sub.n], the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations.
Namely, the languages [U.sub.n] meet the upper bounds for the size of the syntactic semigroup, the quotient complexities of left quotients, the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of the following operations: reversal, star, product (concatenation), and all binary boolean operations.
Also, as shown in [19], if G is an infinite locally compact group and X is any closed subspace of Wap(G) containing [C.sub.0](G), then X is
uncomplemented in [L.sup.[infinity]](G).
