Document Type : Research Paper


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

2 School of Mathematics and Information Science, Shaanxi Normal University


$R\sb{0}$-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.


[1] S. Aguzzoli, M.Busaniche and V. Marra, Spectral duality for nitely generated nilpotent min-
imum algebras with applications, J. Logic Comput., 17 (2007), 749{765.
[2] L. P. Belluce, Semisimple algebras of in nite valued logic and bold fuzzy set theory, Can. J.
Math., 38 (1986), 1356{1379.
[3] W. Blok and D. Pigozzi, Algebraizable logics, Merm. Math. Soc., 77 (1989), 1-89.
[4] M. Botur and A. Dvurecenskij, State-morphism algebras{general approach, Fuzzy Sets Syst.,
218 (2013), 90{102.
[5] M. Busaniche, Free nilpotent minimum algebras, Math. Logic Quart., 52 (2006), 219{236.
[6] C. C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc., 88 (1958),
[7] R. Cignoli and D. Mundici, Stone duality for Dedekind -complete `-groups with order unit,
J. Algebra, 302 (2006), 848{861.
[8] L. C. Ciungu, Non-commutative Multiple-Valued Logic Algebras, Springer, New York, 2014.
[9] L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part I, Arch. Math.
Logic, 52 (2013), 335{376.
[10] L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part II, Arch.
Math. Logic, 52 (2013), 707{732.
[11] A. Di Nola and A. Dvurecenskij, State-morphism MV-algebras, Ann. Pure Appl. Logic, 161
(2009), 161{173.
[12] A. Di Nola, A. Dvurecenskij and A. Lettieri, On varieties of MV-algebras with internal states,
Inter. J. Approx. Reason., 51 (2010), 680{694.
[13] A. Di Nola, A. Dvurecenskij and A. Lettieri, Stone duality type theorems for MV-algebras
with internal states, Comm. Algebra, 40 (2012), 327{342.
[14] A. Dvurecenskij, J. Rachunek and D. Salounova, State operators on generalizations of fuzzy
structures, Fuzzy Sets Syst., 187 (2012), 58{76.
[15] C. Elkan, The paradoxical success of fuzzy logic, IEEE Expert, 9 (1994), 3{8.
[16] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous
t-norms, Fuzzy Sets Syst., 124 (2001), 271{288.
[17] T. Flaminio and F. Montagna, MV-algebras with internal states and probabilistic fuzzy logic,
Inter. J. Approx. Reason., 50 (2009), 138{152.
[18] J. Fodor, Nilpotent minimum and related connectives for fuzzy logic, in: Proc. of the 4th
Inter. Conf. on Fuzzy Syst., March 20-24, Yokohama, (1995), 2077{2082.
[19] P. F. He, X. L. Xin and Y.W. Yang, On state residuated lattices, Soft Comput., 19 (2015),
[20] A. Iorgulescu, Algebras of Logic as BCK-algebras, Editura ASE, Bucarest, 2008.
[21] H. W. Liu and G. J. Wang, Uni ed forms of fully implication restriction methods for fuzzy
reasoning, Inf. Sci., 177(3) (2007), 956{966.
[22] L. Liu and K. Li, Involutive monoidal t-norm based logic and R0-logic, Inter. J. Intelligent
Syst., 199 (2004), 491{497.
[23] Y. M. Liu and M. K. Luo, Fuzzy topology, World Scienti c, Hong Kong, (1997), 15{68.
[24] D. Mundici, Advanced  Lukasiewicz Calculus and MV-algebras, Springer, New York, (2011),
[25] D. Mundici, Averaging the truth-value in  Lukasiewicz sentential logic, Stud. Logica, 55
(1995), 113{127.
[26] Z. M. Ma and Z. W. Fu, Algebraic study to generalized Bosbach states on residuated lattices,
Soft Comput., 19 (2015), 2541{2550.
[27] D. W. Pei, On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets Syst., 138
(2003), 187{195.
[28] D. W. Pei, R0-implication: characteristics and applications, Fuzzy Sets Syst., 131 (2002),
[29] D. W. Pei and G. J. Wang, The completeness and applications of the formal system L, Sci.
China F, 45 (2002), 40{50.
[30] M. H. Stone, The theory of representation for Boolean algebras, Trans. Amer. Math. Soc.,
40 (1936), 37{111.
[31] G. J. Wang, A formal deductive system for fuzzy propositional calculus, Chin. Sci. Bull., 42
(1997), 1521{1526.
[32] G. J. Wang, Fuzzy logic and fuzzy reasoning, In: Proc. of the 7th National Many-Valued and
Fuzzy Logic Conf., November 10-13, Xi'an, (1996), 82{96.
[33] G. J. Wang, Implication lattices and their fuzzy implication space representation theorem,
Acta Math. Sin., (in Chinese), 42 (1999), 133{140.
[34] G. J. Wang, L-Fuzzy Topological Spaces, Shaanxi Normal Univ. Press, Xi'an, (in Chinese),
(1988), 18{56.
[35] G. J. Wang, X. J. Hui and J. S. Song, The R0-type fuzzy logic metric space and an algorithm
for solving fuzzy modus ponens, Comput. Math. Appl., 55(9) (2008), 1974{1987.
[36] G. J. Wang and H. J. Zhou, Introduction to Mathematical Logic and Resolution Principle,
Science Press, Beijing, (2009), 156{298.
[37] S. M. Wang, B. S. Wang and X. Y. Wang, A characterization of truth-functions in the
nilpotent minimum logic, Fuzzy Sets Syst., 145 (2004), 253{266.
[38] D. X. Zhang and Y. M. Liu, L-fuzzy version of Stone's representation theorem for distributive
lattices, Fuzzy Sets Syst., 76 (1995), 259{270.
[39] H. J. Zhou, Probabilistically Quantitative Logic and its Applications, Science Press, Beijing,
(in Chinese), 2015.
[40] H. J. Zhou, G. J. Wang and W. Zhou, Consistency degrees of theories and methods of graded
reasoning in n-valued R0-logic (NM-logic), Inter. J. Approx. Reason., 43 (2006), 117{132.
[41] H. J. Zhou and B. Zhao, Characterizations of maximal consistent theories in the formal
deductive system L (NM-logic) and Cantor space, Fuzzy Set Syst., 158 (2007), 2591{2604.
[42] H. J. Zhou and B. Zhao, Generalized Bosbach and Riecan states based on relative negations
in residuated lattices, Fuzzy Sets Syst., 187 (2012) 33-57.
[43] H. J. Zhou and B. Zhao, Stone-like representation theorems and three-valued lters in R0-
algebras (nilpotent minimum algebras), Fuzzy Sets Syst., 162 (2011), 1{26.