Completeness theory for the product of finite partial algebras Original Research Article

A general completeness criterion for the finite product ∏P(ki) of full partial clones P(ki) (composition-closed subsets of partial operations) defined on finite sets E(ki) (|E(ki)|⩾2, i=1,…,n, n⩾2) is considered and a Galois connection between the lattice of subclones of ∏P(ki), called partial n-clones, and the lattice of subalgebras of multiple-base invariant relation algebra, with operations of a restricted quantifier free calculus, is established. This is used to obtain the full description of all maximal partial n-clones via multiple-base invariant relations and, thus, to solve the general completeness problem in ∏P(ki).