Generalized Marshall Quotients and Real Semigroups of Continuous and Differentiable Functions

The theory of real semigroups developed by M.

Dickmann and A.

Petrovich provides an algebraic framework for abstract real spectra and real algebraic geometry, yet its application to rings of continuous functions is historically hindered by the topological constraints.

In this paper, we bridge this gap by introducing generalized Marshall quotients over rings of real-valued continuous and differentiable functions, yielding new explicitly calculated examples of real semigroups.

Furthermore, we conclude that the group of invertible elements of these quotients constitutes a real reduced hyperfield (which is categorically equivalent to reduced special groups), addressing the open problem of characterizing when the units of a real semigroup form a reduced special group.

Finally, we apply this hyperalgebraic machinery to translate topological and differential phenomena into hyperalgebraic identities, establishing generalized versions of the Łojasiewicz-type inequalities.