The next step of the word problem over monoids

abstract It is known that a group presentationP can be regarded as a 2-complex with a single 0-cell.Thus we can consider a 3-complex with a single 0-cell which is known as a 3-presentation.Similarly, we can also consider 3-presentations for monoids. In this paper, by usingspher-ical monoid pictures, we show that there exists a “nite 3-monoid-presentation which hasunsolvable generalized identity problem that can be thought as the next step (or one-dimension higher) of the word problem for monoids. We note that the method used in thispaper has chemical and physical applications. 2011 Elsevier Inc. All rights reserved. 1. IntroductionPresentations of algebras through “nite rewriting systems have received much attention since they admit simple algo-rithms for solving the word problem. Especially, when the considered algebras are monoids or groups, the notion of rewrit-ing systems coincides with string-rewriting systems. We may refer to[2,4,10] for the relationship between homological“niteness condition FP