Involutive symmetric Gödel spaces, their algebraic duals and logic

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