STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES

Document Type: Research Paper

Authors

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.

Keywords


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