Minimum bases for equational theories of groups and rings: the work of Alfred Tarski and Thomas Green

Suppose that T is an equational theory of groups or of rings. If T is finitely axiomatizable, then there is a least number μ so that T can be axiomatized by μ equations. This μ can depend on the operation symbols that occur in T . In the 1960’s, Alfred Tarski and Thomas C. Green completely determined the values of μ for arbitrary equational theories of groups and of rings. While Tarski and Green announced the results of their collaboration in 1970, the only fuller publication of their work occurred as part of a seminar led by Tarski at Berkeley during the 1968-69 academic year. The present paper gives a full account of their findings and their proofs. 1. Equational logic and equational theories of algebras Equational logic can be viewed as that fragment of first-order logic in which the only sentences are universal sentences whose quantifier-free part is an equation between terms. The familiar distributive law ∀x∀y∀z[x · (y + z) ≈ x · y + x · z] is an example of such a sentence. Equational logic has no logical connectives, no relation symbols apart from the logical equality symbol ≈, and its sole quantifier is the universal quantifier which plays such a restrained role that it is usually surpressed—the distributive law, for example, is expressed simply as x · (y + z) ≈ x · y + x · z. In comparison to first-order logic, equational logic is equipped with an apparently meager means of expression. Still, many classes of algebras that have found important places in mathematics can be specified by means of equations and certainly reasoning about equations is ubiquitous. Equational logic can be developed as a seperate formal system. Garrett Birkhoff [1935] proved a completeness theorem for equational logic using a system of simple rules of inference referring only to equations. He also proved that the classes of algebras axiomatized by equations are exactly those which are closed with respect to the formation of homomorphic images, subalgebras, and arbitrary direct products—the earliest preservation theorem. Two formalisms for equational logic can differ only in their operation symbols. While for more general considerations arbitrary systems of operation symbols are appropriate, in this paper we restrict our attention to those equational formalisms provided with only systems of finitely many operation symbols. Suppose a formalism has been specified by selecting a system of operation symbols. We will say that a set T of equations is an equational theory if and only if it is closed under logical consequence. A set Σ of equations is a base for T provided T is the set of all logical consequences of Σ. Thus Σ is a set of equational axioms for T . We say that the equational theory T if finitely based if it has a finite base. A set Σ of equations is irredundant if and only if Σ is not logically equivalent to any of its proper subsets. In practice, equational theories arise in two ways: as the set of consequences of some particular set Σ of equations, and as the set of all equations true in all the algebras belonging to some class