Involutive symmetric Gödel spaces, their algebraic duals and logic

Abstract

AbstractIt is introduced a new algebra$$(A, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$(A,⊗,⊕,∗,⇀,0,1)called$$L_PG$$LPG-algebra if$$(A, \otimes , \oplus , *, 0, 1)$$(A,⊗,⊕,∗,0,1)is$$L_P$$LP-algebra (i.e. an algebra from the variety generated by perfectMV-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$$L_PG$$LPG-algebra$$(A, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$(A,⊗,⊕,∗,⇀,0,1)is isomorphic to the lattice of Skolem filters (i.e. special type ofMV-filters) of theMV-algebra$$(A, \otimes , \oplus , *, 0, 1)$$(A,⊗,⊕,∗,0,1). The variety$$\mathbf {L_PG}$$LPGof$$L_PG$$LPG-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 ChangMV-algebra. Any$$L_PG$$LPG-algebra is bi-Heyting algebra. The set of theorems of the logic$$L_PG$$LPGis recursively enumerable. Moreover, we describe finitely generated free$$L_PG$$LPG-algebras.