Dualities and algebraic geometry of Baire functions in non-classical logic

Abstract

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.