Relationships between generalized Bosbach states and L-filters on residuated lattices

Generalized Bosbach states and filters on residuated lattices have been extensively studied in the literature. In this paper, relationships between generalized Bosbach states and residuated-lattice-valued filters, also called L-filters, on residuated lattices are investigated. Particularly, type I and type II L-filters and their subclasses are defined, and some their properties are obtained. Then relationships between special types of L-filters and the generalized Bosbach states are considered where generalized Bosbach states are characterized by some type I or type II L-filters with additional conditions. Associated with these relationships, new subclasses of generalized Bosbach states such as implicative type IV, V, VI states, fantastic type IV states and Boolean type IV states are introduced, and the relationships between various types of generalized Bosbach states are investigated in detail. In particular, the existence of several generalized Bosbach states is provided and, as application, some typical subclasses of residuated lattices such as Rl-monoids, Heyting algebras and Boolean algebras are characterized by these generalized Bosbach states.

[1]  Esko Turunen,et al.  States on semi-divisible generalized residuated lattices reduce to states on MV-algebras , 2008, Fuzzy Sets Syst..

[2]  Chris Cornelis,et al.  Filters of residuated lattices and triangle algebras , 2010, Inf. Sci..

[3]  Anatolij Dvurecenskij,et al.  State operators on generalizations of fuzzy structures , 2012, Fuzzy Sets Syst..

[4]  Esko Turunen,et al.  States on semi-divisible residuated lattices , 2008, Soft Comput..

[5]  Liu Lianzhen,et al.  States on finite monoidal t-norm based algebras , 2011 .

[6]  Li Kaitai,et al.  Fuzzy Boolean and positive implicative filters of BL-algebras , 2005 .

[7]  Vilém Vychodil,et al.  Fuzzy equational logic , 2005, Arch. Math. Log..

[8]  Yang Xu,et al.  On filter theory of residuated lattices , 2010, Inf. Sci..

[9]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[10]  Bin Zhao,et al.  Generalized Bosbach and Riečan states on nucleus-based-Glivenko residuated lattices , 2013, Arch. Math. Log..

[11]  Young Bae Jun,et al.  Fuzzy filters of MTL-algebras , 2005, Inf. Sci..

[12]  Zunwei Fu,et al.  Algebraic study to generalized Bosbach states on residuated lattices , 2015, Soft Comput..

[13]  Franco Montagna,et al.  An Algebraic Approach to States on MV-algebras , 2007, EUSFLAT Conf..

[14]  Anatolij Dvurecenskij,et al.  State-morphism algebras - General approach , 2011, Fuzzy Sets Syst..

[15]  Dumitru Busneag,et al.  A new approach for classification of filters in residuated lattices , 2015, Fuzzy Sets Syst..

[16]  Radomír Halas,et al.  States on commutative basic algebras , 2012, Fuzzy Sets Syst..

[17]  Wieslaw A. Dudek,et al.  Filter theory of BL algebras , 2007, Soft Comput..

[18]  Bin Zhao,et al.  Generalized Bosbach and Riečan states based on relative negations in residuated lattices , 2012, Fuzzy Sets Syst..

[19]  George Georgescu,et al.  Bosbach states on fuzzy structures , 2004, Soft Comput..

[20]  Masoud Haveshki,et al.  A note on some types of filters in MTL-algebras , 2014, Fuzzy Sets Syst..

[21]  Franco Montagna,et al.  MV-algebras with internal states and probabilistic fuzzy logics , 2009, Int. J. Approx. Reason..

[22]  Jiří Rachůnek,et al.  Negation in bounded commutative DRℓ-monoids , 2006 .

[23]  DANIELE MUNDICI,et al.  Averaging the truth-value in Łukasiewicz logic , 1995, Stud Logica.

[24]  Esfandiar Eslami,et al.  Some types of filters in BL algebras , 2006, Soft Comput..

[25]  Lianzhen Liu,et al.  On the existence of states on MTL-algebras , 2013, Inf. Sci..

[26]  Tomás Kroupa,et al.  Representation and extension of states on MV-algebras , 2006, Arch. Math. Log..

[27]  Anatolij Dvurecenskij,et al.  State BCK-algebras and state-morphism BCK-algebras , 2013, Fuzzy Sets Syst..

[28]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[29]  Catalin Busneag,et al.  States on Hilbert Algebras , 2010, Stud Logica.

[30]  Martin Vita,et al.  Fuzzy t-filters and their properties , 2014, Fuzzy Sets Syst..

[31]  E. Turunen.  Mathematics Behind Fuzzy Logic , 1999 .

[32]  Sylvia Pulmannová,et al.  MV-pairs and state operators , 2015, Fuzzy Sets Syst..

[33]  M. H. Burton,et al.  The Relationship Between Various Filter Notions on aGL-Monoid , 1999 .

[34]  Anatolij Dvurecenskij,et al.  Probabilistic Averaging in Bounded Rℓ-Monoids , 2006 .

[35]  Zhen Ming Ma.  Two types of MTL-L-filters in residuated lattices , 2014, J. Intell. Fuzzy Syst..

[36]  Sylvia Pulmannova,et al.  Effect algebras with state operator , 2014, TACL.

[37]  Bao Qing Hu,et al.  Characterizations and new subclasses of I-filters in residuated lattices , 2014, Fuzzy Sets Syst..

[38]  Anatolij Dvurecenskij,et al.  Erratum to "State operators on generalizations of fuzzy structures" [Fuzzy Sets Syst. 187(2012) 58-76] , 2012, Fuzzy Sets Syst..

[39]  Anatolij Dvurečenskij,et al.  On Riečan and Bosbach states for bounded non-commutative $R\ell $-monoids , 2006 .

[40]  George Georgescu,et al.  Mathematical Logic Generalized Bosbach states : Part II Lavinia , 2013 .

[41]  Kaitai Li,et al.  Fuzzy Boolean and positive implicative filters of BL-algebras , 2005, Fuzzy Sets Syst..

[42]  Rajab Ali Borzooei,et al.  Some types of filters in MTL-algebras , 2012, Fuzzy Sets Syst..

[43]  Young Bae Jun,et al.  Intersection-Soft Filters in -Algebras , 2013 .

[44]  Jialu Zhang,et al.  Topological properties of prime filters in MTL-algebras and fuzzy set representations for MTL-algebras , 2011, Fuzzy Sets Syst..

[45]  Zhang Xiangyang,et al.  States on R 0 algebras , 2008, SOCO 2008.

[46]  Anatolij Dvurecenskij,et al.  States on Pseudo MV-Algebras , 2001, Stud Logica.

[47]  Tomás Kroupa,et al.  Every state on semisimple MV-algebra is integral , 2006, Fuzzy Sets Syst..

[48]  Anatolij Dvurecenskij,et al.  On varieties of MV-algebras with internal states , 2010, Int. J. Approx. Reason..

[49]  George Georgescu,et al.  Generalized Bosbach States , 2010, 1007.2575.

[50]  Qingli Da,et al.  A decision tree solution considering the decision maker's attitude , 2005, Fuzzy Sets Syst..

[51]  Bin Zhao,et al.  Stone-like representation theorems and three-valued filters in R0- algebras (nilpotent minimum algebras) , 2011, Fuzzy Sets Syst..

[52]  Anatolij Dvurecenskij,et al.  Bounded commutative residuated ℓ-monoids with general comparability and states , 2006, Soft Comput..

[53]  Anatolij Dvurecenskij,et al.  State morphism MV-algebras , 2011, Int. J. Approx. Reason..

[54]  Jirí Rachunek,et al.  State operators on GMV algebras , 2011, Soft Comput..

[55]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[56]  A. Rezaei,et al.  STATES ON PSEUDO-BCK ALGEBRAS , 2008 .

[57]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[58]  Anatolij Dvurecenskij,et al.  State BL-algebras , 2009, Soft Comput..

[59]  Anatolij Dvurecenskij,et al.  Every Linear Pseudo BL-Algebra Admits a State , 2007, Soft Comput..

[60]  A. Dvurecenskij.  States on Quantum Structures Versus Integrals , 2011 .