** Abelian Algebras and Varieties **

Keith A. Kearnes

Department of Mathematical Sciences

University of Arkansas

An algebra is said to be *abelian* if it satisfies
all sentences of the form

for all **x**, **y**, **u**, **v**
(*t*(**x**,**u**) = *t*(**x**,**v**) implies
*t*(**y**,**u**) = *t*(**y**,**v**))
where *t* is a term.
In our talk we discuss recent results about the
structure of abelian algebras and
of locally finite varieties where all members
are abelian.

In the first part of our talk we will discuss
the relationship between abelian, affine and quasi-affine
algebras. We characterize affine algebras
as precisely those abelian algebras which
generate a variety satisfying
an idempotent Mal'cev condition which fails
to hold in the variety of semilattices.

In the second part of the talk we discuss the
local structure of algebras in
locally finite abelian varieties, and
we explain why such varieties must be finitely
generated. We give an example showing that
locally finite abelian varieties
may be nonfinitely based, but we go on to show
that they cannot be inherently nonfinitely
based.