Kalman structures derived from implicative very true MV-algebras

Abstract

The category 𝕄𝕍∙, whose objects are MV∙-algebras, is the image of the category 𝕄𝕍, whose objects are MV-algebras, under the categorical equivalence induced by the Kalman functor 𝐊. The main aim of this paper is to lift this equivalence to the level of the category 𝕄𝕍ivt, whose objects are implicative very true MV-algebras, and the category 𝕄𝕍∙ivt∙ , whose objects are implicative very true MV∙ivt∙ -algebras. Based on this, we then prove that the logic ̀∙ivt∙ of implicative very true MV∙ivt∙ -algebras is complete and semilinear. In this paper, we first introduce the notions of implicative very true operators and their duals on MV-algebras. The resulting classes of algebras will be called MV𝑖𝑣𝑡-algebras and MV𝑠𝑣𝑓 -algebras, respectively, and we discuss the relations between MV𝑖𝑣𝑡-algebras and other previously introduced related structures. Subsequently, we introduce the notion of MV∙𝑖𝑣𝑡∙ -algebras and show that the classes of MV𝑖𝑣𝑡-algebras and MV∙𝑖𝑣𝑡∙ -algebras are in one-to-one correspondence. We further prove a categorical correspondence between the categories 𝕄𝕍ivt and 𝕄𝕍∙ivt∙ via the functor 𝐊, which is motivated by an old construction of Kalman. Finally, we define the logic ̀∙𝑖𝑣𝑡∙ , whose Lindenbaum–Tarski algebra is an MV∙𝑖𝑣𝑡∙ -algebra, and prove that it is complete and semilinear.