A duality for topological semirings of continuous functions. (English. Russian original) Zbl 0885.46025

Vechtomov, E. M.; Smirnova, M. N.

Russ. Math. Surv. 51, No. 3, 571-572 (1996); translation from Usp. Mat. Nauk 51, No. 3, 187-188 (1996).

The authors consider topological semigroups of the form \(C_p(X,S)\), where \(S\) is a topological semiring with \(0\) and \(1\), \(0\neq 1\), and \(X\) is an \(S\)-Tikhonov space. For certain semirings \(S\) it is established that the category of the semirings \(C_p(X,S)\) with continuous homomorphism preserving constants is equivalent to the category of the corresponding pairs \((X,S)\). This equivalence generalizes a number of results obtained earlier by Kaplansky, Nagata and Shirota.

Reviewer: M.G.Tkachenko (Mexico)

MSC:

46E25	Rings and algebras of continuous, differentiable or analytic functions
54C40	Algebraic properties of function spaces in general topology
54H13	Topological fields, rings, etc. (topological aspects)

Keywords:

semiring; topological lattice; topological semigroups; \(S\)-Tikhonov space