Involutive symmetric Gödel spaces, their algebraic duals and logic

Abstract

It is introduced a new algebra \((A, \otimes , \oplus , *, \rightharpoonup , 0, 1)\) called \(L_PG\)-algebra if \((A, \otimes , \oplus , *, 0, 1)\) is \(L_P\)-algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and \((A,\rightharpoonup , 0, 1)\) is a Gödel algebra (i.e. Heyting algebra satisfying the identity \((x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1)\). The lattice of congruences of an \(L_PG\) -algebra \((A, \otimes , \oplus , *, \rightharpoonup , 0, 1)\) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra \((A, \otimes , \oplus , *, 0, 1)\). The variety \(\mathbf {L_PG}\) of \(L_PG\) -algebras is generated by the algebras \((C, \otimes , \oplus , *, \rightharpoonup , 0, 1)\) where \((C, \otimes , \oplus , *, 0, 1)\) is Chang MV-algebra. Any \(L_PG\) -algebra is bi-Heyting algebra. The set of theorems of the logic \(L_PG\) is recursively enumerable. Moreover, we describe finitely generated free \(L_PG\)-algebras.