Gődel filters in residuated lattices

In this paper, in the spirit of [4], we study a new type of filters in residuated lattices : Gődel filters. So, we characterize the filters for which the quotient algebra that is constructed via these filters is a Gődel algebra and we establish the connections between these filters and other types of filters. Using Gődel filters we characterize the residuated lattices which are Gődel algebras. Also, we prove that a residuated lattice is a Gődel algebra (divisible residuated lattice, MTL algebra, BL algebra) if and only if every filter is a Gődel filter (divisible filter, MTL filter, BL filter). Finally, we present some results about injective Gődel algebras showing that complete Boolean algebras are injective objects in the category of Gődel algebras.

Very true operators on MTL-algebras

The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.