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Hemirings, congruences and the Hewitt realcompactification. (English) Zbl 0831.54023
It is known that the Hewitt realcompactification $$vX$$ of a Tikhonov space $$X$$ is considered as the space of all maximal $$z$$-ideals of the ring $$C(X)$$ with the hull-kernel topology. In this paper the authors describe $$vX$$ using the hemiring $$C_+(X)$$ instead of $$C(X)$$. A hemiring is a non-empty set $$R$$ with two operations ‘+’ and ‘$$\cdot$$’ satisfying all the axioms of a ring except the one that requires the existence of additive inverse and satisfying the additional axiom that $$a \cdot 0 = 0\cdot a = 0$$ for each $$a \in R$$. The set $$C_+(X)$$ of all non-negative real-valued continuous functions on a topological space $$X$$ is a hemiring with respect to the usual sum and multiplication. The authors define equivalence relations, called $$z$$-congruence and real $$z$$-congruence, on $$C_+(X)$$ and make a systematic study of them. Those definitions are too long for this review. For a $$z$$-congruence $$\rho$$ on $$C_+ (X)$$, let $$E(\rho) = \{E(f,g) : (f,g) \in \rho\}$$, where $$E(f,g) = \{x \in X : f(x) = g(x)\}$$. They show that the map $$\rho \to E(\rho)$$ is a bijection from the set of all $$z$$-congruences on $$C_+(X)$$ to the set of all $$z$$- filters on $$X$$. For a Tikhonov space $$X$$, let $$W_R(X)$$ be the set of all maximal real $$z$$-congruences on $$C_+(X)$$. The hull kernel topology on $$W_R(X)$$ is the one generated from a base $$\{W(f,g) : f, g \in C_+ (X)\}$$, where $$W(f,g) = \{\rho \in W_R(X) : (f,g) \in \rho\}$$, for closed sets. Finally, they prove that the space $$W_R(X)$$ with the hull-kernel topology is $$vX$$.

##### MSC:
 54D60 Realcompactness and realcompactification 54C45 $$C$$- and $$C^*$$-embedding 54C30 Real-valued functions in general topology
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