By the word problem for some class of algebraic structures we mean the problem of determining, given a finite set E of equations between words (i.e. terms) and an additional equation x = y, whether x = y must hold in all structures satisfying each member of E. In 1947 Post [P] showed the word problem for semigroups to be undecidable. This result was strengthened in 1950 by Turing, who showed the word problem to be undecidable for cancellation semigroups,i.e. semigroups satisfying the cancellation property
Novikov [N] eventually showed the word problem for groups to be undecidable.
(Many flaws in Turing's proof were corrected by Boone [B]. Even after his corrections, at least one problem remains; the sentence on line 16 of p. 502 of [T] does not follow if one relation is principal and the other is a commutation relation. A corrected and somewhat simplified version of Turing's proof can be built on the construction given here.)
In 1966 Gurevich [G] showed the word problem to be undecidable for finite semigroups. However, this result on finite structures has not been extended to cancellation semigroups or groups; indeed it is easy to see that a finite cancellation semigroup is a group, so both questions are the same. We do not here settle the word problem for finite groups, but we do show that the word problem is undecidable for finite semigroups with zero (that is, having an element 0 such that x0 = 0x = 0 for all x) satisfying an approximation to the cancellation property (1).