A finite basis problem in loop theory
by
Tomasz Kowalski
University of Cagliari

A loop can be thought of as a group-like object, with a nonassociative multiplication. A loop L is a group if and only if L is associative. One effect of nonassociativity of L is that powers of an element a are not well-defined (a^3 is ambiguous between (a*a)*a and a*(a*a), say). So not every one-generated subloop of L is a group. Consider the class of all loops such that every one-generated subloop is a group. Such loops are called power-associative, and form a non-finitely based variety, as shown by Evans and Neumann in [1]. They also considered diassociative loops: loops such that every two-generated subloop is a group, and asked whether these are finitely based. A negative solution was proposed in [2] but found wanting. In [3] the problem was still reported as open "although everyone believes that the solution is in the negative". A recent [4] proposed (the jury is still out) another negative solution relying on some deep results in number theory.

We will show that diassociativity is not finitely based even relative to power associativity. Our result is stronger than the one announced in [4] and the proof technique is different: we use elementary constructions and a standard ultraproduct argument.

References

[1] T. Evans, B.H. Neumann, On varieties of groupoids and loops, J. London Math. Soc. 28 (1953) 342-350.

[2] D.M. Clark, Diassociative groupoids are not finitely based, J. Australian Math. Soc. 11 (1970) 113-114.

[3] M.K. Kinyon, K. Kunen, J.D. Phillips, A generalisation of Moufang and Steiner loops, Algebra Universalis 48, 1 (2002) 81-101.

[4] W.D. Smith, Loop diassociativity has no finite basis, unpublished manuscript (2005).

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Date received: June 8, 2008