  There is a polymodal provability logic $GLP$. We consider generalizations of
this logic: the logics $GLP_{\alpha}$, where $\alpha$ ranges over linear
ordered sets and play the role of the set of indexes of modalities. We consider
the varieties of modal algebras that corresponds to the polymodal logics. We
prove that the elementary theories of the free $\emptyset$-generated $GLP_{n}$
-algebras are decidable for all finite ordinals $n$.
