Characterizations and new subclasses of (I)-filters in residuated lattices

摘要

Filters play an important role in studying logical systems and the related algebraic structures. Various filters have been proposed in the literature. In this paper, we aim to develop a unifying definition for some specific filters called I-filters which provide us with a meaningful method to study these filters and corresponding logical algebras. In particular, trivial characterizations of I-filters, non-trivial characterizations of classes of I-filters, such as implicative, fantastic and Boolean filters, and characterizations of homologous logical algebras are obtained. Next, three new types of I-filters named divisible filters, strong and n-contractive filters in residuated lattices are introduced. Particularly, it is verified that n-fold implicative BL-algebras and n-contractive BL-algebras coincide. Finally, we investigate the relationships between these specific I-filters. It is shown that a filter is a fantastic filter if and only if it is both a divisible filter and a regular filter.