The semiring of continous $[0,1]$-valued functions

In this paper, we study the properties of prime ideals in semirings of continuous functions with values in the unit interval $[0,1]$ on topological spaces. We describe maximal and pure ideals of such semirings. We study homomorphisms of semirings of continuous $[0,1]$-valued functions. In terms of semirings of functions we characterize some properties of compacta. We show that the theory of ideals in these semirings differs from the case of rings of continuous functions.