FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1998, VOLUME 4, NUMBER 2, PAGES 493-510

Semirings of continuous nonnegative functions: divisibility, ideals, congruences

V. I. Varankina
E. M. Vechtomov
I. A. Semenova

Abstract

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Authors investigate the properties of divisibility (GCD, LCM, to be Bezout semiring) in semirings of continuous nonnegative real-valued functions on a topological space $X$. The correspondences between the lattice of ideals of the ring $C(X)$ and the lattice of ideals of the semiring $C+(X)$ are considered. New characterizations of $F$-spaces are obtained. Congruences on abstract semirings are studied. Maximal congruences of semirings $C+(X)$ are described. It is shown that all congruences on a semifield $U(X)$ of all continuous pozitive functions on $X$ are ideal congruences if and only if $X$ is the pseudocompact space.

All articles are published in Russian.

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Last modified: June 17, 1998