Multiplicatively collapsing and rewritable algebras

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.