In this paper we aim at completing the study of $\sigma $-complete Riesz MV-algebras that started in Di Nola et al. (2018, J. Logic Comput., 28, 1275–1292). To do so, we discuss polynomials, algebraic geometry and dualities in the infinitary variety of such algebras. In particular, we characterize the free objects as algebras of Baire-measurable functions and we generalize two dualities, namely the Marra–Spada duality and the Gelfand duality, obtaining a duality with basically disconnected compact Hausdorff spaces and an equivalence with Rickart $C^*$-algebras.

中文翻译：

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非经典逻辑中贝尔函数的对偶性和代数几何

在本文中，我们的目标是完成从 Di Nola 等人开始的对 $\sigma $-complete Riesz MV-algebras 的研究。(2018, J. Logic Comput., 28, 1275–1292)。为此，我们讨论了此类代数的无穷变体中的多项式、代数几何和对偶。特别是，我们将自由对象表征为 Baire 可测函数的代数，并且我们推广了两个对偶，即 Marra-Spada 对偶和 Gelfand 对偶，获得了具有基本不连贯紧致 Hausdorff 空间的对偶和与 Rickart $C^* 的等价性$-代数。