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Multiplicatively collapsing and rewritable algebras


Authors: Eric Jespers, David Riley and Mayada Shahada
Journal: Proc. Amer. Math. Soc. 143 (2015), 4223-4236
MSC (2010): Primary 16R99, 17B30, 20M25
DOI: https://doi.org/10.1090/S0002-9939-2015-12563-4
Published electronically: March 18, 2015
MathSciNet review: 3373922
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Abstract: A semigroup $ S$ is called $ n$-collapsing if, for every $ a_1,\ldots , a_n$ in $ S$, there exist functions $ f\neq g$ (depending on $ a_1,\ldots , a_n$), such that

$\displaystyle a_{f(1)}\cdots a_{f(n)} = a_{g(1)}\cdots a_{g(n)};$

it is called collapsing if it is $ n$-collapsing, for some $ n$. More specifically, $ S$ is called $ n$-rewritable if $ f$ and $ g$ can be taken to be permutations; $ S$ is called rewritable if it is $ n$-rewritable for some $ n$. Semple and Shalev extended Zelmanov's solution of the restricted Burnside problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent. In this paper, we consider when the multiplicative semigroup of an associative algebra is collapsing; in particular, we prove the following conditions are equivalent, for all unital algebras $ A$ over an infinite field: the multiplicative semigroup of $ A$ is collapsing, $ A$ satisfies a multiplicative semigroup identity, and $ A$ satisfies an Engel identity. We deduce that, if the multiplicative semigroup of $ A$ is rewritable, then $ A$ must be commutative.

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Additional Information

Eric Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: efjesper@vub.ac.be

David Riley
Affiliation: Department of Mathematics, Western University, London, Ontario, Canada N6A 5B7
Email: dmriley@uwo.ca

Mayada Shahada
Affiliation: Department of Mathematics, Western University, London, Ontario, Canada N6A 5B7
Email: mshahada@uwo.ca

DOI: https://doi.org/10.1090/S0002-9939-2015-12563-4
Keywords: Multiplicative semigroup, adjoint semigroup, collapsing, rewritable, semigroup identity, Lie nilpotent, Engel identity
Received by editor(s): March 28, 2014
Received by editor(s) in revised form: June 12, 2014
Published electronically: March 18, 2015
Additional Notes: The authors acknowledge support from Onderzoeksraad of Vrije Universiteit, Fonds voor Wetenschappelijk Onderzoek (Vlaanderen) and NSERC of Canada.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society