Universal Algebra

The typical feature of a mathematical theory is that it deals with collections or sets of objects, where certain relations exist between the objects of these sets, or between different sets, while the nature of the objects is entirely immaterial. A simplification can be achieved by considering only objects which are themselves sets. At first sight this appears to lead to a vicious circle, but the difficulty may be resolved by beginning with the empty set. On the other hand it is necessary to restrict the sets which may appear as members of other sets, if one wants to avoid the contradictions arising from the consideration of ‘the set of all those sets which are not members of themselves’ (Russell’s paradox). One therefore introduces a different term, such as ‘class’, for general collections of objects, and distinguishes those classes which are themselves members of other classes by calling them sets. Without entering fully into the question of axiomatics here (for which the reader may be referred to more detailed accounts such as Bourbaki [54], Gödel [40], Kelley [55], and Wang & McNaughton [53]), we shall give a list of axioms, which in the main express the conditions under which a class is to be regarded as a set.

Many important classes of algebras, among them groups, rings, and lattices, consist of all the homomorphic images of certain ‘free’ algebras in the class, which are essentially determined by the cardinal of a free generating set. These are the varieties of algebras, which form the subject of Chapter IV, but free algebras are also of importance in more general situations, and we therefore devote a chapter to the study of properties of free algebras which are independent of the notion of a variety. A free algebra is itself a special case of the notion of a universal functor in category theory, and so we shall first describe universal functors in general categories (cf. Samuel [48], MacLane [63]).