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Mathematical Problems in Engineering
Volume 2013, Article ID 908623, 6 pages
Research Article

A Bijection between Lattice-Valued Filters and Lattice-Valued Congruences in Residuated Lattices

1School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China
2College of Computer Science and Technology, Taiyuan University of Technology, Taiyuan 030024, China

Received 22 January 2013; Accepted 1 July 2013

Academic Editor: Bin Liu

Copyright © 2013 Wei Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The aim of this paper is to study relations between lattice-valued filters and lattice-valued congruences in residuated lattices. We introduce a new definition of congruences which just depends on the meet and the residuum . Then it is shown that each of these congruences is automatically a universal-algebra-congruence. Also, lattice-valued filters and lattice-valued congruences are studied, and it is shown that there is a one-to-one correspondence between the set of all (lattice-valued) filters and the set of all (lattice-valued) congruences.