Soft Computing

, Volume 22, Issue 7, pp 2321–2328 | Cite as

Fuzzy extended filters on residuated lattices

Methodologies and Application


The notion of fuzzy extended filters is introduced on residuated lattices, and its essential properties are investigated. By defining an operator \(\rightsquigarrow \) between two arbitrary fuzzy filters in terms of fuzzy extended filters, two results are immediately obtained. We show that (1) the class of all fuzzy filters on a residuated lattice forms a complete Heyting algebra, and its classical version is equivalent to the one introduced in Kondo (Soft Comput 18(3):427–432, 2014), which is defined with respect to (crisp) generated filters of singleton sets; (2) the connection between fuzzy extended filters and fuzzy generated filters is built, with which three other classes generating complete Heyting algebras, respectively, are presented. Finally, by the aid of fuzzy t-filters, we also develop the characterization theorems of the special algebras and quotient algebras via fuzzy extended filters.


Complete Heyting algebra Fuzzy extended filters Fuzzy generated filters Fuzzy t-filters Residuated lattice 



This work is supported by the National Natural Science Foundation of China (Nos. 11371130; 11101135) and Research Fund for the Doctoral Program of Higher Education of China (No. 20120161110017).

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.

Ethical approval

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


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaChina

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