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Undecidability of the identity problem for finite semigroups

Published online by Cambridge University Press:  12 March 2014

Douglas Albert ,
Robert Baldinger  and
John Rhodes
Douglas Albert
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Robert Baldinger
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, E-mail: [email protected]
John Rhodes
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, E-mail: [email protected]
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Extract

In 1947 E. Post [28] and A. A. Markov [18] independently proved the undecidability of the word problem (or the problem of deducibility of relations) for semigroups. In 1968 V. L. Murskil [23] proved the undecidability of the identity problem (or the problem of deducibility of identities) in semigroups.

If we slightly generalize the statement of these results we can state many related results in the literature and state our new results proved here. Let V denote either a (Birkhoff) variety of semigroups or groups or a pseudovariety of finite semigroups. By a very well-known theorem a (Birkhoff) variety is defined by equations or equivalently closed under substructure, surmorphisms and all products; see [7]. It is also well known that V is a pseudovariety of finite semigroups iff V is closed under substructure, surmorphism and finite products, or, equivalently, determined eventually by equations w1 = w1′, w2 = w2′, w3 = w3′,… (where the finite semigroup S eventually satisfies these equations iff there exists an n, depending on S, such that S satisfies Wj = Wj′ for j ≥ n). See [8] and [29]. All semigroups form a variety while all finite semigroups form a pseudovariety.

We now consider a table (see the next page). In it, for example, the box denoting the “word” (identity) problem for the psuedovariety V” means, given a finite set of relations (identities) E and a relation (identity) u = ν, the problem of whether it is decidable that E implies u = ν inside V.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 1 , March 1992 , pp. 179 - 192
Copyright
Copyright © Association for Symbolic Logic 1992

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References

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