Closed ideals and closed congruences of semirings of $[0,1]$-valued functions with topology of pointwise convergence

For an arbitrary Tychonoff space $X$, we describe closed ideals and closed congruences of the topological semiring $C_p(X,\mathbf I)$ of all continuous functions on $X$ with values in the closed unit interval $\mathbf I$ considered in the topology of pointwise convergence. The duality between the category of Tychonoff spaces $X$ with continuous mappings and the category of topological semirings $C_p(X,\mathbf I)$ with continuous homomorphisms preserving constants is established.