Barker C. C. H.. Some calculations in logic. The mathematical gazette, vol. 41 (1957), pp. 108–111.

in the set of laws of the other. To this is then easily reduced the comparison of finite algebras which are not of the same species and also of those whose operators are not uniform. The main result may be stated as follows. If m is the product of the numbers of elements of two finite algebras of the same species then there is a number l0 m £ 1 + 2 TM' i = l such that the two algebras have the same set of laws if and only if every equation, ip = ip\ where the number of different primitive symbols does not exceed m and the ranks of y> and of y>' are smaller than ta , is either a law of both or of none of them. First, using the fact that the laws of both algebras are just those of the direct product, it is shown (theorem 1) that if every equation where the number of different primitive symbols does not exceed m is a law of both algebras or of none of them, then the same happens to any equation whatsoever. Next, t0 is shown to exist, bounded as above, with the property that for every function of rank ^ t0 and number of primitive symbols ^ m there is a function of rank < /„ and the same primitive symbols which has also the same sequences of values obtained by substitution of the elements of the direct product for the primitive symbols (theorem 2). Finally, the problem is pointed out of extending the results to algebras of infinite order. Here Reseller's comment in XVIII 268 seems to be applicable.