STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES

$Rsb{0}$-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 $Rsb{0}$-algebras and the category of three-valued Stone spaces.Then we extend Flaminio-Montagna internal states to $Rsb{0}$-algebras.Such internal states must be idempotent MV-endomorphisms of $Rsb{0}$-algebras.Lastly we present a Stone duality for the category of MV-skeletons of $Rsb{0}$-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 $Rsb{0}$-algebras.