Multiplicative semigroups of continuous functions

Multiplicative bijections of semigroups of interval-valued continuous functions

“…Finally we remark that our results are essentially different from those given in [5] for multiplicative bijections between spaces C(X, R) of real-valued continuous functions on X. Even so, there is a similarity in that the multiplicative structure of the spaces of functions determines the space X up to homeomorphism.…”

On the ideal structure of certain semirings and compactification of topological spaces

“…These semirings include various rings of continuous functions and the biregular rings (with identity) of Arens and Kaplansky [2], in addition to 7?-lattices. The notion of i?-ideal is a generalization of the notions of lattice ideal and O-ideal of Milgram [12]. The present paper and [16] seem to overlap very little, except in some of the applications.…”

“…The following definition is derived from Milgram's notion of O-ideal [12] and that of lattice ideal [3]. Definition 1.2.…”

On the equivalence of the ring, lattice, and semigroup of continuous functions

“…Under the assumption that X is a compact Hausdorff space, it was' shown in 1937 by M. H. Stone [8] and in 1939 by Gelfand and Kolmogoroff [2] that C(X) determines X. In 1947 and 1949, under the same hypothesis, Kaplansky [4] and Milgram [5] showed respectively that each of L(X), SeX) determine X. Generalizing these results, Hewitt [3] in 1948 and Shirota [7] in 1952 showed respectively that if X is a Q-space (for definition, see [3 ]…”