On comparison of finite algebras
Jan Kalicki · Proceedings of the American Mathematical Society · 1952
Introduction. In this paper we shall describe an effective procedure to determine in a finite number of steps whether two abstract algebras A' and of finite order have: (i) the same set of laws, (ii) the set of laws of one of them is included in the set of laws of the other, (iii) distinct sets of laws, (iv) overlapping sets of laws. The procedure is a generalization of certain results obtained by the author in case of truth-tables.' Some parts of the paper On the structure of finite algebras by G. Birkhoff2 will be presupposed, namely: (i) the definition of an abstract algebra A = (T, F) and of its order (?2, pp. 433-434), (ii) the definition of the direct product of abstract algebras (?7, pp. 437-438), (iii) the definition of an abstract algebra of species 1, and of a uniform operator (?8), (iv) the definition of a function 4 of rank n associated with the speciesl, (?9, Definition 2), (v) the idea of a substitution t of elements of an algebra for each primitive symbol of a function 4 and of the resulting value t(4) of the function (?9, p. 439), (vi) the definition of a law of an algebra (?9, Definition 3), (vii) the idea of a law of a set of algebras (?9, p. 439), (viii) the theorem that the set of laws of any aggregate of algebras A is the same as that of the direct product A of all the A (Corollary 2, p. 440). The notation used by Birkhoff in the paper referred to will be followed without further explanation. L(Ao) will stand for the set of laws of any algebra Ao.