Soft Computing

, Volume 23, Issue 23, pp 12233–12240 | Cite as

Two examples of subalgebras of the set of functions between bounded lattices

  • Zhi-qiang Liu
  • Xue-ping WangEmail author


This paper deals with the convolution operations on the set of functions between bounded lattices. Firstly, we show a subset of the set of functions that is both a bisemilattice and a Birkhoff system under the convolution operations and then show another subset of the set of functions that is a bounded distributive lattice under the convolution operations, while the domain lattice of the functions does not need to be distributive.


Convolution operations Bisemilattice Birkhoff system Distributive lattice 



The authors thank the referees for their valuable comments and suggestions.

Compliance with ethical standards

Conflict of interest

All the authors in the paper have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Birkhoff G (1967) Lattice theory, vol 25, 3rd edn. Colloquium Publishers, American Mathematical Society, ProvidencezbMATHGoogle Scholar
  2. De Miguel L, Bustince H, De Baets B (2018) Convolution lattices. Fuzzy Sets Syst 335:67–93MathSciNetCrossRefGoogle Scholar
  3. Harding J, Walker C (2018) A topos view of the type-2 fuzzy truth value algebra. arXiv:1810.07565
  4. Harding J, Walker C, Walker E (2016) The truth value algebra of type-2 fuzzy sets. CRC Press, Boca RatonCrossRefGoogle Scholar
  5. Harding J, Walker C, Walker E (2018) The convolution algebra. Algebra Univers 79:33. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Inf Control 31:312–340MathSciNetCrossRefGoogle Scholar
  7. Nieminen J (1977) On the algebraic structure of fuzzy sets of type-2. Kybernetika 13:261–273MathSciNetzbMATHGoogle Scholar
  8. Walker C, Walker E (2005) The algebra of fuzzy truth values. Fuzzy Sets Syst 149:309–347MathSciNetCrossRefGoogle Scholar
  9. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8:199–249MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesSichuan Normal UniversityChengduPeople’s Republic of China

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