Soft Computing

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

Generalized state operators on residuated lattices

  • Michiro KondoEmail author


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 



This study was funded by JSPS KAKENHI (Grant Number 15K00024).

Compliance with ethical standards

Conflicts of interest

The author declares that he has no conflict of interest.


  1. Ciungu LC (2008) Bosbach and Riečan states on residuated lattices. J Appl Funct Anal 3:175–188MathSciNetzbMATHGoogle Scholar
  2. Constantinescu NM (2014) State filters on fuzzy structures with internal states. Soft Comput 18:1841–1852CrossRefzbMATHGoogle Scholar
  3. Constantinescu NM (2012) On pseudo-BL algebras with internal state. Soft Comput 16:1915–1922CrossRefzbMATHGoogle Scholar
  4. Dvurečenskij A (2001) States on pseudo MV-algebras. Studia Log 68:301–327MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dvurečenskij A, Rachůnek J (2006) Probabilistic averaging in bounded R\({\ell }\)-monoids. Semigroup Forum 72:190–206MathSciNetzbMATHGoogle Scholar
  6. Dvurečenskij A, Rachůnek J (2006) On Riečan and Bosbach states for bounded non-commutative R\({\ell }\)-monoids. Math Slovaca 56:487–500MathSciNetzbMATHGoogle Scholar
  7. Dvurěcenskij A, Rachůnek J, Šalounova D (2012) State operators on generalizations of fuzzy structures. Fuzzy Sets Syst 187:58–76MathSciNetCrossRefzbMATHGoogle Scholar
  8. Flaminio T, Montanga F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50:138–152MathSciNetCrossRefzbMATHGoogle Scholar
  9. Georgescu G (2004) Bosbach states on fuzzy structures. Soft Comput 8:217–230CrossRefzbMATHGoogle 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, AmsterdamzbMATHGoogle Scholar
  11. Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  12. Haveshki M, Mohamadhasani M (2012) Extended filters in bounded commutative R\(\ell \)-monoids. Soft Comput 16:2165–2173CrossRefzbMATHGoogle Scholar
  13. Hart JB, Rafter J, Tsinakis C (2002) The structure of commutative residuated lattices. Int J Algebra Comput 12:509–524MathSciNetCrossRefzbMATHGoogle Scholar
  14. Haveshki M, Saeid AB, Eslami E (2006) Some types of filters in BL algebras. Soft Comput 10:657–664CrossRefzbMATHGoogle Scholar
  15. He P, Xin X, Yang Y (2015) On state residuated lattices. Soft Comput 19:2083–2094CrossRefzbMATHGoogle Scholar
  16. Kondo M (2014) Characterization of extended filters in residuated lattices. Soft Comput 18:427–432CrossRefzbMATHGoogle Scholar
  17. Kondo M (2014) States on bounded commutative residuated lattices. Math Slovaca 64:1093–1104MathSciNetCrossRefzbMATHGoogle 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 44:21–34MathSciNetzbMATHGoogle Scholar
  21. Rachůnek J, Salounova D (2011) State operators on GMV-algebras. Soft Comput 15:327–334CrossRefzbMATHGoogle Scholar
  22. Turunen E (2001) Boolean deductive systems of BL-algebras. Arch Math Logic 40:467–473MathSciNetCrossRefzbMATHGoogle Scholar
  23. Ward M, Dilworth RP (1939) Residuated lattices. Trans AMS 45:335–354MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Tokyo Denki UniversityTokyoJapan

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