Which Finite Algebras are Finitely Based?
by
George F. McNulty
University of South Carolina

Ralph McKenzie's resolution of Tarski's Finite Basis Problem tells us that there is no algorithm for determining which among all finite algebras with finitely many fundamental operations have finitely based equational theories. We must settle instead for widely and easily applicable sufficient conditions and for such necessary conditions. This talk will take up sufficient conditions.

In the literature, apart from some apparently ad hoc results, there seem to be four general directions: that direction loosely based on investigation of the clone of term operations of the algebra (e.g. work of Lyndon, Murskii, and Berman); that direction leading to the conclusion that each finite group and each finite ring is finitely based (e.g. work of Oates and Powell and of L'vov and of Kruse); that direction launched by Ralph McKenzie's proof that finite lattices with operators are finitely based which led, through work of Kiby Baker, to Ross Willard's proof concerning finite algebras which generate residually finite congruence meet-semidistributive varieties; and finally that direction launched, again by McKenzie, centered on the role of definable principal congruences and leading to recent results of Kirby Baker and Ju Wang.

The bulk of the talk will address advances and prospects for unifying the last three directions. The advances are collaborative work with Kirby Baker and Ju Wang. The talk will conclude with a number of open problems.

Date received: February 8, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-70.