Algebra universalis

, 79:83 | Cite as

Kites and residuated lattices

  • Michal Botur
  • Anatolij DvurečenskijEmail author


We investigate a construction of an integral residuated lattice starting from an integral residuated lattice and two sets with an injective mapping from one set into the second one. The resulting algebra has a shape of a Chinese cascade kite, therefore, we call this algebra simply a kite. We describe subdirectly irreducible kites and we classify them. We show that the variety of integral residuated lattices generated by kites is generated by all finite-dimensional kites. In particular, we describe some homomorphisms among kites.


Residuated lattice Kite algebra Subdirect irreducible kite Classification of kites 

Mathematics Subject Classification

03G10 03B50 



The authors are very indebted to an anonymous referee for his/her careful reading and suggestions which helped us to improve the presentation of the paper.


  1. 1.
    Aglianò, P., Montagna, F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181, 105–129 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13, 437–461 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Di Nola, A., Georgescu, G., Iorgulescu, G.: Pseudo-BL algebras I. Multi. Valued Log. 8, 673–714 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Di Nola, A., Georgescu, G., Iorgulescu, G.: Pseudo-BL algebras II. Multi. Valued Log. 8, 715–750 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dvurečenskij, A.: Pseudo MV-algebras are intervals in \(\ell \)-groups. J. Aust. Math. Soc. 72, 427–445 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dvurečenskij, A.: Aglianò–Montagna type decomposition of linear pseudo hoops and its applications. J. Pure Appl. Algebra 211, 851–861 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dvurečenskij, A., Giuntini, R., Kowalski, T.: On the structure of pseudo BL-algebras and pseudo hoops in quantum logics. Found. Phys. 40, 1519–1542 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dvurečenskij, A., Kowalski, T.: Kites and pseudo BL-algebras. Algebra Univers. 71, 235–260 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Galatos, N., Tsinakis, C.: Generalized MV-algebras. J. Algebra 283, 254–291 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras. Mult. Valued Log. 6, 95–135 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Glass, A.M.W.: Partially Ordered Groups. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  13. 13.
    Hájek, P.: Basic fuzzy logic and BL-algebras. Soft Comput. 2, 124–128 (1998)CrossRefGoogle Scholar
  14. 14.
    Jipsen, P., Montagna, F.: On the structure of generalized BL-algebras. Algebra Univers. 55, 226–237 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mundici, D.: Interpretations of \(AF\,C^{\star }\)-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rachůnek, J.: A non-commutative generalization of MV-algebras. Czechoslov. Math. J. 52, 255–273 (2002)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Palacký University Olomouc Faculty of SciencesOlomoucCzech Republic
  2. 2.Mathematical Institute Slovak Academy of SciencesBratislavaSlovakia

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