z-congruences and topologies on \(C^+(X)\)

For a Tychonoff space X, \(C^+(X)\) $C^{+}(X)$ denotes the non-negative real-valued continuous functions on X. We obtain a correlation between z-congruences on the ring C(X) and z-congruences on the semiring \(C^+(X)\) $C^{+}(X)$ . We give a new characterization of P-spaces via z-congruences on \(C^+(X)\) $C^{+}(X)$ . The z-congruences on \(C^+(X)\) $C^{+}(X)$ are shown to have an algebraic nature like z-ideals. We study some topological properties of \(C^+(X)\) $C^{+}(X)$ under u-topology and m-topology. It is shown that a proper ideal of \(C^+(X)\) $C^{+}(X)$ is closed under m-topology if and only if it is the intersection of maximal ideals of \(C^+(X)\) $C^{+}(X)$ . Also, we prove that every ideal of \(C^+(X)\) $C^{+}(X)$ is closed if and only if X is a P-space. We investigate the connectedness and compactness of \(C^+(X)\) $C^{+}(X)$ under m-topology. It is shown that the component of \(\varvec{0}\) $\mathbf{0}$ is \(C_\psi (X)\cap C^+(X)\) $C_{\psi }(X)\cap C^{+}(X)$ . Finally, we show that \(C_m^+(X)\) $C_m^{+}(X)$ is locally compact, \(\sigma \) $\sigma$ -compact and hemicompact if and only if X is finite.