Semi-G-filters, Stonean filters, MTL-filters, divisible filters, BL-filters and regular filters in residuated lattices

At present, the filter theory of BLtextit{-}algebras has been widelystudied, and some important results have been published (see for examplecite{۴}, cite{۵}, cite{xi}, cite{۶}, cite{۷}). In other works such ascite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory inthe more general setting of residuated lattices is done, generalizing thatfor BLtextit{-}algebras. Note that filters are also characterized byvarious types of fuzzy sets. Most of such characterizations is trivial butsome are nontrivial, for example characterizations obtained in cite{xm}.Both situation have revealed a rich range of classes of filters: Boolean,implicative, Heyting, positive implicative, fantastic (or MV-filter), etc.In this paper we work in the general cases of residuated lattices and put inevidence new types of filters in a residuated lattice (in the spirit of cite{mvl}): semi-G-filterstextit{, }Stonean filters, divisible filters,BL-filters and regular filters.

Keywords:

residuated lattice , Boolean algebra , BL-algebra , MV-algebra , MTL-algebra , Divisible residuated lattice , Regular residuated lattice , Deductive system , Filter , Boolean filter , MTL-filter , Divisible filter , BL-filter , MV-filter , Semi-G-filter , Stonean filter , Regular filter