STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES

R0-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 R0-algebras and the category of three-valued Stone spaces. Then we extend Flaminio-Montagna internal states to R0-algebras. Such internal states must be idempotent MVendomorphisms of R0-algebras. Lastly we present a Stone duality for the category of MV-skeletons of R0-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 R0-algebras.