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 \(vX\).


54D60 Realcompactness and realcompactification
54C45 \(C\)- and \(C^*\)-embedding
54C30 Real-valued functions in general topology
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