广义马歇尔商与连续可微函数的实半群
Generalized Marshall Quotients and Real Semigroups of Continuous and Differentiable Functions
AI总结：
本文针对实半群理论在连续函数环应用中的拓扑约束问题，引入广义马歇尔商，给出实半群新示例，得出商的可逆元群构成实约化超域，解决相关开放问题，并应用超代数机制建立广义不等式。
摘要：
M. Dickmann和A. Petrovich发展的实半群理论为抽象实谱和实代数几何提供了代数框架，但其在连续函数环上的应用受拓扑约束阻碍。本文通过引入实值连续可微函数环上的广义马歇尔商弥合这一差距，给出实半群新的显式计算示例。还得出这些商的可逆元群构成实约化超域，解决了实半群单位何时构成约化特殊群的开放问题。最后应用此超代数机制将拓扑和微分现象转化为超代数恒等式，建立了洛雅西维茨型不等式的广义版本。
英文摘要：
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.
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