MULTIPLICATIVELY COLLAPSING AND REWRITABLE ALGEBRAS

作者：Jespers Eric^*; Riley David; Shahada Mayada

来源：Proceedings of the American Mathematical Society, 2015, 143(10): 4223-4236.

DOI：10.1090/s0002-9939-2015-12563-4

• Multiplicative semigroup
• adjoint semigroup
• collapsing
• rewritable
• semigroup identity
• Lie nilpotent
• Engel identity

A semigroup S is called n-collapsing if, for every a(1),..., a(n) in S, there exist functions f not equal g (depending on a(1),..., a(n)), such that
a(f(1))...a(f(n)) = a(g(1))...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.