u-Isomorphic semigroups of continuous functions

Abstract

A. CSASZAR (Budapest), member of the Academy 0. Introduction. Let X be a topological space, and denote by C(X) the set of all real-valued continuous functions defined on X, by C*(X) the subset of C(X) composed of bounded functions. Both C(X) and C*(X) can be considered as a ring under pointwise addition and multiplication of functions, or as a semigroup under pointwise multiplication. For a completely regular Hausdorff space X, let fix and vX denote the (~ech Stone compactification and the Hewitt realcompactification of X, respectively (see e. g.[2]).The following propositions are well-known for completely regular Hausdortf spaces X and Y:A. If C(X) and C(Y) are ring isomorphic, then vX and vY are homeomorphic. B. If C*(X) and C* (Y) are ring isomorphic, then fix and fly are homeomorphic.It is also known that, both in Propositions A and B, ring isomorphy can be replaced by semigroup isomorphy. Moreover, for A, an essentially stronger result is valid. In order to formulate it, let us recall (cf.[1]) that, in a semigroup S, we write at>b for a, bCS, iffthere exists an element c~S such that a=cb; for two semigroups $1 and S~ and the respective relations ~>i and t>~, a bijection ~p: SI-*S~ is said to be a d-isomorphism iff a~,-l b ~, q~(a)~>~q~(b).C. (ef.[1], Theorem 3). For two completely regular Hausdorff spaces X and Y, if C(X) and C(Y) are d-isomorphic, then vX and vY are homeomorphic. ~

Homomorphisms on Lattices of Continuous Functions

“…It is clear that any multiplicative bijection on C(X, (0, 1)) does. We refer the reader to the papers [22,24,14,8] for related results on the semigroup C(X).…”