Advertisement

Abstract

The pseudo-hoops were originally introduced by Bosbach under the name of complementary semigroups. It was proved that a pseudo-hoop has the pseudo-divisibility condition and it is a meet-semilattice, so a bounded R-monoid can be viewed as a bounded pseudohoop together with the join-semilattice property. In other words, a bounded pseudohoop is a meet-semilattice ordered residuated, integral and divisible monoid.

In this chapter we present the main notions and results regarding the pseudo-hoops, we prove new properties of these structures, we introduce the notions of join-center and cancellative-center of pseudo-hoops and we define and study algebras on subintervals of pseudo-hoops.

References

  1. 1.
    Aglianò, P., Ferreirim, I.M.A., Montagna, F.: Basic hoops: an algebraic study of continuous t-norms. Stud. Log. 87, 73–98 (2007) zbMATHCrossRefGoogle Scholar
  2. 10.
    Blok, W.J., Ferreirim, I.M.A.: On the structure of hoops. Algebra Univers. 43, 233–257 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 15.
    Bosbach, B.: Komplementäre Halbgruppen. Axiomatik und Aritmetik. Fundam. Math. 64, 257–287 (1969) MathSciNetzbMATHGoogle Scholar
  4. 16.
    Bosbach, B.: Komplementäre Halbgruppen. Kongruenzen und Quotienten. Fundam. Math. 69, 1–14 (1970) MathSciNetzbMATHGoogle Scholar
  5. 37.
    Chajda, I., Kühr, J.: A note on interval MV-algebra. Math. Slovaca 56, 47–52 (2006) MathSciNetzbMATHGoogle Scholar
  6. 55.
    Ciungu, L.C.: Directly indecomposable residuated lattices. Iran. J. Fuzzy Syst. 6, 7–18 (2009) MathSciNetzbMATHGoogle Scholar
  7. 105.
    Dvurečenskij, A., Hyčko, M.: Algebras on subintervals of BL-algebras, pseudo-BL algebras and bounded residuated R-monoids. Math. Slovaca 56, 125–144 (2006) MathSciNetzbMATHGoogle Scholar
  8. 106.
    Dvurečenskij, A., Kowalski, T.: On decomposition of pseudo-BL algebras. Math. Slovaca 61, 307–326 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 148.
    Georgescu, G., Leuştean, L., Preoteasa, V.: Pseudo-hoops. J. Mult.-Valued Log. Soft Comput. 11, 153–184 (2005) zbMATHGoogle Scholar
  10. 195.
    Jakubík, J.: On intervals and the dual of a pseudo-MV algebra. Math. Slovaca 56, 213–221 (2006) MathSciNetzbMATHGoogle Scholar
  11. 196.
    Jakubík, J.: On interval subalgebras of generalized MV-algebras. Math. Slovaca 56, 387–395 (2006) MathSciNetzbMATHGoogle Scholar
  12. 198.
    Jipsen, P.: An overview of generalized basic logic algebras. Neural Netw. World 13, 491–500 (2003) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Lavinia Corina Ciungu
    • 1
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA

Personalised recommendations