We define a class, denoted APP, of real-valued functions f : {0, 1}n → [0, 1] such that f can be approximated to within any > 0 by a probabilistic Turing machine running in time poly(n, 1/ ). The class APP can be viewed as a generalization of BPP. We argue that APP is more natural and more important than BPP, and that most results about BPP are better stated as results about APP. We show that A...

The lattice of z-ideals of the ring C(X) of real-valued continuous functions on a completely regular Hausdorff space X has been shown by Mart́ınez and Zenk to be a complete Heyting algebra with certain properties. We show that these properties are due only to the fact that C(X) is an f -ring with bounded inversion. This we do by studying lattices of algebraic z-ideals of abstract f -rings with b...

Let S be a semigroup. In certain cases we give some characterizations of extreme amenability of S and we show that in these cases extreme left amenability and extreme right amenability of S are equivalent. Also when S is a compact topological semigroup, we characterize extremely left amenable subalgebras of C(S), where C(S) is the space of all continuous bounded real valued functions on S