Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups

Abstract

We use additive notation, +, 0, ',-, for Abelian groups, and multiplicative notation, , e, l, /, for ordinary groups. Throughout this note, and / are binary operations rather than abbreviations for, e.g., x + y' and x Xy-1 One might think it trivial, given (2), to obtain a single axiom in terms of product and inverse, by simply rewriting a/,(3 to a * l Doing so gives a single axiom, but then * is not product, and -1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and the identity element [8]. Single equational axioms in terms of product and inverse have been reported by Neumann [5] and others [3, 2]. In this note we consider single amplicative axioms, that is, axioms of the form ct = ,X3 y = b. For Abelian groups, an axiom of this type with five variables was