Simple Word Problems in Universal Algebras’

An algor i thm is descr ibed which is capable of solving cer ta in word problems: i .e . of deciding whether or not two words composed of var iables and operators can be proved equal as a consequence of a given set of identities satisf ied by the opera tors . Al though the genera l word problem is wel l 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 opera tor -, and the nul lary operator e , the a lgor i thm is capable of deducing f rom the three identitiesa.(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 .ais not a consequence of the given axioms. The method is based on a well-ordering of the set of all words, such that each identity can be construed as a “reduction”, in the sense that the right-hand side of the identity represents a word smaller in the ordering than the left-hand side. A set of reduction identities is said to be “complete” when two words are equal as a consequence of the ident i t ies i f and only i f they reduce to the same word by a series of reductions. The method used in this algorithm is essentially to test whether a given set of identities is complete; if it is not complete the algorithm in many cases finds a new consequence of the identities which can be added to the l i s t . The process i s repeated unt i l e i ther a comple te se t i s achieved or unt i l an anomalous situation occurs which cannot at present be handled. Results of several computational experiments using the algorithm are given.