Boolean filters and positive implicative filters of residuated lattices

The aim of this paper is to introduce the notions of Boolean filters and positive implicative filters in residuated lattices and to investigate their properties. Several characterizations of Boolean filters and positive implicative filters are derived. The extension theorems of implicative filters and positive implicative filters are obtained. The relations among Boolean filters, implicative filters and positive implicative filters are investigated and it is proved that Boolean filters are equivalent to implicative filters, and that every Boolean filter is a positive implicative filter, but the converse may not be true. Furthermore, the conditions under which a positive implicative filter is a Boolean filter are established. (C) 2007 Elsevier Inc. All rights reserved.