The completeness problem on the product of algebras of finite-valued logic

Gives a general completeness criterion for the arity-calibrated product P_kxP_m of the algebras of all functions of the k-valued and m-valued logics (k,m⩾2). The Galois connection between the lattice of subalgebras P_kxP_m and the lattice of subalgebras of the double-base invariant relations algebra (with operations of restricted first order calculus) is established. This is used to obtain a Slupecki type criterion for P_kxP_m and to solve the completeness problem in P _kxP_m (m⩾2)