An algebraic approach to unification under associativity and commutativity

From the work of Siekmann & Livesey, and Stickel it is known how to unify two terms in an associative and commutative theory: transfer the terms into Abelian strings, look for mappings which solve the problem in the Abelian monoid, and decide whether a mapping can be regarded as a unifier. Very often most of the mappings are thus eliminated, and so it is crucial for efficiency either to not create these unnecessary solutions or to remove them as soon as possible. The following work formalises the transformations between the free algebra and this monoid. This leads to an algorithm which uses maximal information for its search for solutions in the monoid. It is both very efficient and easily verifiable. Some applications of this algorithm are shown in the appendix