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The Burnside problem for semigroups of matrices. (English) Zbl 0564.20045
Combinatorics on words. Progress and perspectives, Proc. Int. Meet., Waterloo/Can. 1982, 279-295 (1983).
[For the entire collection see Zbl 0552.00014.]
The author gives new proofs of results stating the finiteness of torsion subsemigroups of matrix- or PI-rings: theorems of Brown, McNaughton- Zalcstein, Jacob. He gives an extension of Brown’s result to the ”strongly locally finite” case, and proves that every torsion subsemigroup of a PI-ring is locally finite, a result which is stated without proof in the paper of McNaughton-Zalcstein. The approach is combinatorial, and uses Ramsey-type theorems of T. C. Brown and Shirshov (the first one is proved in the paper). The author gives the following open problem: is every torsion semigroup of matrices over a commutative semiring finite?
Reviewer: Ch.Reutenauer

MSC:
20M25 Semigroup rings, multiplicative semigroups of rings
16Rxx Rings with polynomial identity
16S50 Endomorphism rings; matrix rings