Simple Word Problems in Universal Algebras

Abstract

An algorithm is described which is capable of solving certain word problems: i.e. of deciding whether or not two words composed of variables and operators can be proved equal as a consequence of a given set of identities satisfied by the operators. Although the general word problem is well known to be unsolvable, this algorithm provides results in many interesting cases. For example in elementary group theory if we are given the binary operator ·, the unary operator −, and the nullary operator e, the algorithm is capable of deducing from the three identities a · (b · c) = (a · b) · c, a · a− = e, a · e = a, the laws a− · a = e, e · a = a, a– = a, etc.; and furthermore it can show that a · b = b · a− is not a consequence of the given axioms.