Extended Ideals in residuated lattices

Document Type : Original Article


1 Science School, Xi'an Polytechnic University, Xi'an, China

2 department mathematics, northwest university, xian, china


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. 


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