Involutive symmetric Godel spaces, their algebraic duals and logic

It is introduced a new algebra (A,circle times, circle plus, * , ->, 0, 1) called L(P)G-algebra if(A,circle times, circle plus, * 0, 1) is LP-algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and (A, ->, 0, 1) is a Godel algebra (i.e. Heyting algebra satisfying the identity (x -> y) v (y -> x) = 1). The lattice of congruences of an L(P)G-algebra (A, circle times, circle plus, * ->, 0, 1) is isomorphic to the lattice of Skolem filters (i.e. special type of M V-filters) of the M V-algebra (A, circle times, circle plus, * 0, 1). The variety LPG of LPG-algebras is generated by the algebras (C, circle times, circle plus, * ->, 0, 1) where (C, circle times, circle plus, *, 0, 1) is Chang MV-algebra. Any L(P)G-algebra is bi-Heyting algebra. The set of theorems of the logic L(P)G is recursively enumerable. Moreover, we describe finitely generated free L(P)G-algebras.