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Abstract

Filter theory plays a vital role in studying fuzzy logical systems and their algebraic structures. Various types of filters have been extensively investigated in the literature. In this paper, several types of filters related to the Stonean axiom in residuated lattices are introduced such as Stonean filters, subimplicative filters, De Morgan filters, RDP-filters and (weak) subimplicative-Stonean filters, the relationships between them and some existing types of filters are obtained. Particularly, it is shown that a filter is a weak subimplicative-Stonean filter if and only if subimplicative filter coincides with Stonean filter. By these filters, the Stonean axiom and its alternative definitions in an arbitrary residuated lattice are investigated. In particular, it is proved that Stonean axiom and subimplicative axiom coincide when the residuated lattice is subimplicative-Stonean, specially, De Morgan or RDP.

Keywords

Non-classical logics residuated lattice Stonean filter subimplicative filter De Morgan filter RDP-filter (weak) subimplicative-Stonean filter

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References

Busneag

and Piciu

, Some types of filters in residuated lattices, Soft Computing 18 (2014), 825–837.

Busneag

and Piciu

, A new approach for classification of filters in residuated lattices, Fuzzy Sets and Systems 260 (2015), 121–130.

Borumand Saeid

and Zahiri

, Radicals in MTL-algebras, Fuzzy Sets and Systems 236 (2014), 91–103.

Borzooei

R.A.

, Shoar

S.K.

and Ameri

, Some types of filters in MTL-algebras, Fuzzy Sets and Systems 187 (2012), 92–102.

Bahls

, Cole

, Galatos

, Jipsen

and Tsinakis

, Cancellative residuated lattices, Algebra Universalis 50 (2003), 83–106.

Castano

, Diaz Varela

J.P.

and Torrens

, Freedecomposability in varieties of pseudocomplemented residuated lattices, Studia Logica 98 (2011), 223–235.

Cattaneo

, Ciucci

, Giuntini

and Konig

, Algebraic structures related to many valued logical systems -part I: Heyting Wajsberg algebras, Fundamenta Informaticae 63 (2004), 331–355.

Cignoli

and Torrens

, Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic, Archive for Mathematical Logic 42 (2003), 361–370.

Cignoli

, Free algebras in varieties of Stonean residuated lattices, Soft Computing 12 (2008), 315–320.

10.

Esteva

and Godo

, Monoidal t-norm based logic: Towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001), 271–288.

11.

Haveshki

, Borumand Saeid

and Eslami

, Some types of filters in BL-algebras, Soft Computing 10 (2006), 657–664.

12.

Haveshki

, A note on some types of filters in MTL-algebras, Fuzzy Sets and Systems 247 (2014), 135–137.

13.

Hájek

, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.

14.

Kondo

and Dudek

W.A.

, Filter theory of BL-algebras, Soft Computing 12 (2008), 419–423.

15.

Kondo

and Turunen

, Prime filters on residuated lattices, IEEE 42nd International Symposium on Multiple-Valued Logic, 2012.

16.

Kondo

, Simple characterization of strict residuated lattices with an involutive negation, Soft Computing 17 (2013), 39–44.

17.

Z.M.

and Hu

B.Q.

, Characterizations and new subclasses of I-filters in residuated lattices, Fuzzy Sets and Systems 247 (2014), 92–107.

18.

Z.M.

, MTL-filters and their characterizations on residuated lattices, Computer Engineering and Applications 48 (2012), 64–66 (in Chinese).

19.

Pei

D.W.

, On the equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy sets and Systems 138 (2003), 187–195.

20.

Turunen

, Mathematics behind Fuzzy Logic, Physica-Verlag, 1999.

21.

Turunen

, BL-algebras of basic fuzzy logic, Mathware and Soft Computing 6 (1999), 49–61.

22.

Turunen

, Boolean deductive systems of BL-algebra, Archive for Mathematical Logic 40 (2001), 467–473.

23.

Van Gasse

, Deschrijver

, Cornelis

and Kerre

E.E.

, Filters of residuated lattices and triangle algebras, Information Sciences 180 (2010), 3006–3020.

24.

Víta

, Why are papers about filters on residuated structures (usually) trivial? Information Sciences 276 (2014), 387–391.

25.

Wang

G.J.

, An Introduction to Mathematical Logic and Resolution Principle, Science in China Press, Beijing, 2003 (in Chinese).

26.

Wang

S.M.

, A fuzzy logic for the revised drastic product t-norm, Soft Computing 11 (2007), 585–590.

27.

Zhang

, Topological properties of prime filters in MTL-algebras and fuzzy set representations for MTL-algebras, Fuzzy Sets and Systems 178 (2011), 38–53.

28.

Zhu

Y.Q.

and Xu

, On filter theory of residuated lattices, Information Sciences 180 (2010), 3614–3632.

29.

Zhou

and Zhao

, Stone-like representation theorems and three-valued filters in R₀-algebras (nilpotent minimum algebras), Fuzzy Sets and Systems 162 (2011), 1–26.

Several types of filters related to the Stonean axiom in residuated lattices

Abstract

Keywords

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