Soft Computing

, Volume 21, Issue 20, pp 6063–6071 | Cite as

Generalized state operators on residuated lattices



We give a definition of a generalized state operator (g-state operator) \(\sigma \) on a residuated lattice X and a g-state residuated lattice \((X,\sigma )\), by which the class of all g-state residuated lattices is proved to be a variety, and consider properties of g-state residuated lattices. We prove some fundamental results about them, such as characterizations of \(\sigma \)-filters, extended \(\sigma \)-filters, homomorphism theorems for g-state residuated lattices. Moreover, we show that every g-state residuated lattice is a subdirect product of subdirectly irreducible g-state residuated lattices.


(Generalized) state filters Extended filters Residuated lattices 


  1. Ciungu LC (2008) Bosbach and Riečan states on residuated lattices. J Appl Funct Anal 3:175–188MathSciNetMATHGoogle Scholar
  2. Constantinescu NM (2014) State filters on fuzzy structures with internal states. Soft Comput 18:1841–1852CrossRefMATHGoogle Scholar
  3. Constantinescu NM (2012) On pseudo-BL algebras with internal state. Soft Comput 16:1915–1922CrossRefMATHGoogle Scholar
  4. Dvurečenskij A (2001) States on pseudo MV-algebras. Studia Log 68:301–327MathSciNetCrossRefMATHGoogle Scholar
  5. Dvurečenskij A, Rachůnek J (2006) Probabilistic averaging in bounded R\({\ell }\)-monoids. Semigroup Forum 72:190–206MathSciNetMATHGoogle Scholar
  6. Dvurečenskij A, Rachůnek J (2006) On Riečan and Bosbach states for bounded non-commutative R\({\ell }\)-monoids. Math Slovaca 56:487–500MathSciNetMATHGoogle Scholar
  7. Dvurěcenskij A, Rachůnek J, Šalounova D (2012) State operators on generalizations of fuzzy structures. Fuzzy Sets Syst 187:58–76MathSciNetCrossRefMATHGoogle Scholar
  8. Flaminio T, Montanga F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50:138–152MathSciNetCrossRefMATHGoogle Scholar
  9. Georgescu G (2004) Bosbach states on fuzzy structures. Soft Comput 8:217–230CrossRefMATHGoogle Scholar
  10. Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics, studies in logic and the foundations of mathematics. Elsevier, AmsterdamMATHGoogle Scholar
  11. Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, DordrechtCrossRefMATHGoogle Scholar
  12. Haveshki M, Mohamadhasani M (2012) Extended filters in bounded commutative R\(\ell \)-monoids. Soft Comput 16:2165–2173CrossRefMATHGoogle Scholar
  13. Hart JB, Rafter J, Tsinakis C (2002) The structure of commutative residuated lattices. Int J Algebra Comput 12:509–524MathSciNetCrossRefMATHGoogle Scholar
  14. Haveshki M, Saeid AB, Eslami E (2006) Some types of filters in BL algebras. Soft Comput 10:657–664CrossRefMATHGoogle Scholar
  15. He P, Xin X, Yang Y (2015) On state residuated lattices. Soft Comput 19:2083–2094CrossRefMATHGoogle Scholar
  16. Kondo M (2014) Characterization of extended filters in residuated lattices. Soft Comput 18:427–432CrossRefMATHGoogle Scholar
  17. Kondo M (2014) States on bounded commutative residuated lattices. Math Slovaca 64:1093–1104MathSciNetCrossRefMATHGoogle Scholar
  18. Kondo M, Kawaguchi MF (2016) Some properties of generalized state operators on residuated lattices. Proc IEEE ISMVL 2016:162–166MathSciNetGoogle Scholar
  19. Kondo M, Watari O, Kawaguchi MF, Miyakoshi M (2007) A logic determined by commutative residuated lattices. EUSFLAT Conference, pp 45–48Google Scholar
  20. Kôpka F, Chovanec F (1994) D-Posets. Math Slovaca 44:21–34MathSciNetMATHGoogle Scholar
  21. Rachůnek J, Salounova D (2011) State operators on GMV-algebras. Soft Comput 15:327–334CrossRefMATHGoogle Scholar
  22. Turunen E (2001) Boolean deductive systems of BL-algebras. Arch Math Logic 40:467–473MathSciNetCrossRefMATHGoogle Scholar
  23. Ward M, Dilworth RP (1939) Residuated lattices. Trans AMS 45:335–354MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Tokyo Denki UniversityTokyoJapan

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