On the variety of Gödel MV-algebras

Abstract

We introduce a new algebraic structure

$$\begin{aligned} (A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1) \end{aligned}$$

called Gödel–MV-algebra (GMV-algebra) such that

• $(A, \otimes , \oplus , *, 0, 1)$ is MV-algebra;

• $(A,\vee , \wedge ,\rightharpoonup , 0, 1)$ is a Gödel algebra (i. e. Heyting algebra satisfying the identity $(x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1$).

It is shown that the lattice of congruences of a GMV -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 $(A, \otimes , \oplus , *, 0, 1)$. Any GMV-algebra is bi-Heyting algebra. Any chain GMV-algebra is simple, and any GMV-algebra is semi-simple. Finitely generated GMV-algebras are described, and finitely generated finitely presented GMV-algebras are characterized. The algebraic counterpart of axiomatically presented GMV-logic is GMV-algebras .