Heyting, Boolean and Pseudo-MV Filters in Residuated Lattices

The paper is devoted to introduce the notions of left-(right-)Heyting, left-(right-)Boolean and left-(right-)MV filter in residuated lattices and to investigate their properties. Several characterizations of these notions are derived. We show that each left-(right-)Heyting filter is left-(right)normal and each left- and right-Heyting filter is a Heyting filter. The relations between left-(right-)Boolean filters and left-(right-)Heyting filters are investigated and we prove that left-(right-)Boolean filters are left-(right-)Heyting and this implication is strict. The conditions under which a left-(right-)Heyting filter is left-(right-)Boolean are established. Finally, we show that a filter is Boolean if and only if it is Heyting and pseudo-MV.