The smallest ideal of the Stone-Cech compactification of N under multiplication
by
Gugu Moche
Howard University

As is the case with any discrete semigroup, the operation + on N can be extended to the Stone-Cech compactification \betaN of N so that (\betaN, +) is a right topological semigroup with N contained in its topological center. (That is, for each p in \betaN, the function \rho_p:\betaN --> \betaN, defined by \rho_p(x)=x + p, is continuous, and for each n in N, the function \lambda_n:\betaN --> \betaN, defined by \lambda_n(x)=n + x, is continuous.) The algebraic structure of (\betaN, +) has been extensively studied, and in particular a great deal is known about the structure of its smallest two sided ideal, K(\betaN, +).

Much less is known about the structure of (\betaN, ·) and its smallest ideal K(\betaN, ·). We investigate this structure using extensions of homomorphisms from (N, ·) to (N, +). In particular, we show that each maximal group in K(\betaN, ·) contains a copy of the free group on 2^c generators.

Date received: June 4, 1999