Polynomial completeness criteria in finite Boolean algebras

For a given finite Boolean algebra with r(r/spl ges/2) atoms we consider the set BF(r) of all polynomials produced by superpositions of the main operations and r atomic constants. Using the isomorphism between BF(r) and P, the arity-calibrated product of r two-valued logic algebras P/sub 2/, and also the description of all maximal subalgebras of P/sub 2//sup r/, we establish a general completeness criterion in BF(r), a Sheffer criterion for a single Boolean function to be a generating element in BF(r), and Slupecki type criterion in BF(r) as well.