STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES

$R\sb{0}$-algebras, which were proved to be equivalent to Esteva and Godo's NM-algebras modelled by Fodor's nilpotent minimum t-norm, are the equivalent algebraic semantics of the left-continuous t-norm based fuzzy logic firstly introduced by Guo-jun Wang in the mid 1990s.
In this paper, we first establish a Stone duality for the category of MV-skeletons of $R\sb{0}$-algebras and the category of three-valued Stone spaces.
Then we extend Flaminio-Montagna internal states to $R\sb{0}$-algebras.
Such internal states must be idempotent MV-endomorphisms of $R\sb{0}$-algebras.
Lastly we present a Stone duality for the category of MV-skeletons of $R\sb{0}$-algebras with Flaminio-Montagna internal states and the category of three-valued Stone spaces with Zadeh type idempotent continuous endofunctions.
These dualities provide a topological viewpoint for better understanding of the algebraic structures of $R\sb{0}$-algebras.