In this paper we consider fundamental properties of some types of filters (Boolean, positive implicative, implicative and fantastic filters) of BL algebras defined in Haveshki et al. (Soft Comput 10:657-664, 2006) and Turunen (Arch Math Logic 40:467-473, 2001). It is proved in Haveshki et al. (2006) that if F is a maximal and (positive) implicative filter then it is a Boolean filter. In that paper there is an open problem Under what condition are Boolean filters positive implicative filters?One of our results gives an answer to the problem, that is, we need no more conditions. Moreover, we give simple characterizations of those filters by an identity form ∀x, y(t (x, y) ∈ F), where t (x, y) is a term containing x, y.
The notion of n-normal residuated lattice, as a class of residuated lattices in which every prime filter contains at most n minimal prime filters, is introduced and studied. Before that, the notion of ω-filter is introduced and it is observed that the set of ω-filters in a residuated lattice forms a distributive lattice on its own, which includes the set of coannulets as a sublattice. The class of n-normal residuated lattices is characterized in terms of their prime filters, minimal prime filters, coannulets and ωfilters. 1 1 2010 Mathematics Subject Classification: 06F99,06D20
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.