%0 Journal Article
%T STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES
%J Iranian Journal of Fuzzy Systems
%I University of Sistan and Baluchestan
%Z 1735-0654
%A Zhou, Hongjun
%A Shi, Hui-Xian
%D 2017
%\ 08/30/2017
%V 14
%N 4
%P 139-161
%! STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES
%K $Rsb{0}$-algebra
%K Nilpotent minimum algebra
%K MV-skeleton
%K internal state
%K Stone duality
%R 10.22111/ijfs.2017.3330
%X $Rsb{0}$-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.
%U http://ijfs.usb.ac.ir/article_3330_f12a0f74c6087b8efa1b0345bf21060c.pdf