The decidability of craig's interpolation property in well-composed $\mathrm J$-logics

Under study are the extensions of Johansson's minimal logic $\mathrm J$. We find sufficient conditions for the finite approximability of $\mathrm J$-logics in dependence on the form of their axioms. Using these conditions, we prove the decidability of Craig's interpolation property (CIP) in well-composed $\mathrm J$-logics. Previously all $\mathrm J$-logics with weak interpolation property (WIP) were described and the decidability of WIP over $\mathrm J$ was proved. Also we establish the decidability of the problem of amalgamability of well-composed varieties of $\mathrm J$-algebras.