Spectral dualities of $MV$-algebras

Abstract

Hong and Nel in [8] obtained a number of spectral dualities between a cartesian closed topological category ${\bf X}$ and a category of algebras of suitable type in ${\bf X}$ in accordance with the original formalism of Porst and Wischnewsky[12]. In this paper, there arises a dual adjointness $S\vdash C$ between the category ${\bf X}={\mathcal L}im$ of limit spaces and that ${\bf A}$ of $MV$-algebras in ${\bf X}.$ We firstly show that the spectral duality: $S({\bf A})^{op}\simeq C({\bf X}^{op})$ holds for the dualizing object $K=I=[0, 1]$ or $K=2=\{0, 1\}.$ Secondly, we study a duality between the category of Tychonoff spaces and the category of semi-simple $MV$-algebras. Furthermore, it is shown that for any $X\in{\mathcal L}im \, (X\neq\emptyset)$ $C(X, I)$ is densely embedded into a cube $I^{|H|}$, where $H$ is a set.