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The word problem for cancellation semigroups with zero

Published online by Cambridge University Press:  12 March 2014

Yuri Gurevich  and
Harry R. Lewis
Yuri Gurevich
Affiliation:
Computer and Communication Sciences Department, University of Michigan, Ann Arbor, Michigan 48109
Harry R. Lewis
Affiliation:
Aiken Computation Laboratory, Harvard University, Cambridge, Massachusetts 02138
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Extract

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).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 1 , March 1984 , pp. 184 - 191
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

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