Multiplicatively collapsing and rewritable algebras
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- by Eric Jespers, David Riley and Mayada Shahada PDF
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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 \[ a_{f(1)}\cdots a_{f(n)} = a_{g(1)}\cdots a_{g(n)};\] 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.References
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Additional Information
- Eric Jespers
- Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
- MR Author ID: 94560
- 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
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3373922