Completeness theory for vector partial multiple-valued logic functions

A general completeness criterion for the finite product ΠP (k_i) of algebras P(k_i) of all partial functions of k_i-valued logic (k_i⩾2, i=1,…n; n⩾2) is considered and a Galois connection between the lattice of subalgebras of ΠP(k_i) and the lattice of subalgebras of multiple-base invariant relations algebra (with operations of a restricted quantifier free calculus) is established. This is used to obtain the full description of all maximal subalgebras of ΠP(k_i) and, thus, to solve the completeness problem in ΠP(k_i)