@article{gurevich1966the,
author = {Gurevich, Yuri},
title = {The Word Problem for Some Classes of Semigroups},
year = {1966},
month = {July},
abstract = {The word problem for finite semigroups is the following decision problem: given some number n of word pairs (u1,v1), ..., (un,vn) and an additional word pair (u0,v0), decide whether the n equations u1=v1,..., un=vn impliy the additional equation u0=v0 in all finte semigroups. We prove that the word problem for finite semigroups is undecidable.

In fact, the undecidability result holds for a particular premise   E   =  (u1=v1 and ... and un=vn). Furthermore, this particular E can be chosen so that the following classes K1 and K2 of word pairs are recursively inseparable:
K1 is the class of word pairs (u0,v0) such that E implies u0=v0 in every periodic semigroup.
K2 is the class of word pairs (u0,v0) such that E fails to imply u0=v0 in some finite semigroup.
The paper contains some additional undecidability results.},
url = {http://approjects.co.za/?big=en-us/research/publication/word-problem-classes-semigroups/},
pages = {25-35},
journal = {Algebra and Logic (Russian)},
}