Extended Ideals in residuated lattices

Abstract

In this paper, we generalized the notion of extended ideals and stable ideals associated to a subset B of a residuated lattices L and we discuss what kind of residuated lattices have extended ideals. Then we investigate the related properties of them. We show that if L is an involutive residuated lattice, then EI(B) is a stable ideal relative to B and So, EI(B)= $\bigcap $ { J : J is stable relative to B and $I \subseteq J $}. We also give a characterization of this extended ideal: EI(B)= (B] --> I in the complete Heyting algebra $(\operatorname{Id}(L), \wedge, \vee, \rightarrow,\{0\}, L) $. We also prove that the class S(B) of all stable ideals relative to $B \subseteq L $ is a complete Heyting algebra. Finally, we prove that the set of extended ideals and the set of extended filters on involutive residuated lattices are one-to-one correspondence.