Positive implicative ordered filters of implicative semigroups

Abstract

We introduce the notion of positive implicative ordered filters in implicative semigroups. We show that every positive implicative ordered filter is both an ordered filter and an implicative ordered filter. We give examples that an ordered filter (an implicative ordered filter) may not be a positive implicative ordered filter. We also give equivalent conditions of positive implicative ordered filters. Finally we establish the extension property for positive implicative ordered filters.

Keywords

• Implicative semigroup
• ordered filter
• (positive) implicative ordered filter.