Interval-valued intuitionistic (T, S)-fuzzy filters theory on residuated lattices

Abstract

The aim of this paper is further to develop the filter theory on residuated lattices. Firstly, the notion of interval valued intuitionistic (T, S)-fuzzy filter(IVI (T, S)-fuzzy filter for short) on residuated lattices is introduced by linking the interval valued intuitionistic fuzzy set, t-norm, s-norm and filter theory of residuated lattices; the properties and equivalent characterizations of interval valued intuitionistic (T, S)-fuzzy filter are investigated; the relation between IVI (T, S)-fuzzy filter and filter is studied. Secondly, the notions of interval valued intuitionistic (T, S)-fuzzy implicative filter and interval valued intuitionistic (T, S)-fuzzy Boolean filter are introduced; the properties and equivalent characterizations of them are investigated; the intuitionistic (T, S)-fuzzy implicative filter is proved to be equivalent to the intuitionistic (T, S)-fuzzy Boolean filter in residuated lattices. Finally, the intuitionistic (T, S)-fuzzy positive implicative filter and intuitionistic (T, S)-fuzzy G (MV) filter are introduced; some equivalent characterizations of them are obtained and the relations among these fuzzy filters are investigated.