FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

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.

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Last modified: June 17, 1998