Zhou, H., Shi, H. (2017). STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES. Iranian Journal of Fuzzy Systems, 14(4), 139-161. doi: 10.22111/ijfs.2017.3330

Hongjun Zhou; Hui-Xian Shi. "STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES". Iranian Journal of Fuzzy Systems, 14, 4, 2017, 139-161. doi: 10.22111/ijfs.2017.3330

Zhou, H., Shi, H. (2017). 'STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES', Iranian Journal of Fuzzy Systems, 14(4), pp. 139-161. doi: 10.22111/ijfs.2017.3330

Zhou, H., Shi, H. STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES. Iranian Journal of Fuzzy Systems, 2017; 14(4): 139-161. doi: 10.22111/ijfs.2017.3330

STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES

^{1}School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, CHINA

^{2}School of Mathematics and Information Science, Shaanxi Normal University

Abstract

$R\sb{0}$-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.

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