Semirings of functions determine finite $T_{o}$ topologies

Abstract:

An analogue of the Stone-Gelfand-Kolmogoroff theorem for compact Hausdorff spaces is proven for finite ${T_0}$ topological spaces. Let $C(X)$ be the semiring of continuous functions from finite ${T_0}$ X into Z, the nonnegative integers with open sets of the form $\{ 0,1,2, \cdots ,m\}$. Products and sums in $C(X)$ are defined pointwise. Denote the set of nonzero semiring homomorphisms of $C(X)$ into Z by $H(X)$ and give it the compact-open topology where $C(X)$ is considered discrete. Then (1) X and $H(X)$ are homeomorphic. (2) $C(X)$ is semiring isomorphic to $C(Y)$ iff X is homeomorphic to Y. (3) The topology of X can be completely recovered from the inclusion relations among the ideals of $C(X)$ which are kernels of the elements in $H(X)$.