Zero-Set Intersection Graph On C+(X)

TL;DR

This paper explores the relationships between zero-set intersection graphs and algebraic structures of function semirings on Tychonoff spaces, establishing equivalences among various isomorphisms and topological properties.

Contribution

It introduces the zero-set intersection graph on C+(X) and proves the equivalence of graph, algebraic, and topological isomorphisms for realcompact spaces.

Findings

Graph isomorphism corresponds to semiring isomorphism.

Algebraic isomorphisms imply topological homeomorphisms.

Results connect graph properties with topological and algebraic structures.

Abstract

For any Tychonoff space X we have introduced the zero-set in-tersection graph on {\Gamma}(C+(X)) and studied the graph properties in connection with the algebraic properties of the semiring C+(X). We have shown that for any two realcompact spaces X and Y the graph isomorphism between {\Gamma}(C+(X)) and {\Gamma}(C+(Y )), the semiring isomorphism between C+(X) and C+(Y ), the topological homeomorphism between X and Y, the ring isomorphism between C(X) and C(Y ) and the graph isomorphism between {\Gamma}(C(X)) and {\Gamma}(C(Y )) are equivalent.