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

Abstract

At present, the filter theory of $BL$textit{-}algebras has been widely<br />studied, and some important results have been published (see for example<br />cite{4}, cite{5}, cite{xi}, cite{6}, cite{7}). In other works such as<br />cite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory in<br />the more general setting of residuated lattices is done, generalizing that<br />for $BL$textit{-}algebras. Note that filters are also characterized by<br />various types of fuzzy sets. Most of such characterizations is trivial but<br />some are nontrivial, for example characterizations obtained in cite{xm}.<br />Both situation have revealed a rich range of classes of filters: Boolean,<br />implicative, Heyting, positive implicative, fantastic (or MV-filter), etc.<br />In this paper we work in the general cases of residuated lattices and put in<br />evidence new types of filters in a residuated lattice (in the spirit of cite<br />{mvl}): semi-G-filterstextit{, }Stonean filters, divisible filters,<br />BL-filters and regular filters.