Survey 1979: Equational Logic - Department of Mathematics ...

WALTER TAYLOR
forms of mathematics from the Theorem of Pythagoras onward). While the maturity
and value of mathematical logic are unquestioned nowadays, we hope the reader will
also gain an insight into the present-day vigor (if not yet maturity) of general algebra.
This subject has been clouded by a skepticism ranging from Marczewski's [283]
sympathetic warning:
[In subjects like general topology and general algebra]
it is easy to get stranded in trivial topics, and caught in
the net of overdetailed conditions, of futile
generalizations.
to the outright malediction: "nobody should specialize in it" ([184], [64]). This
injunction is certainly out of date (if indeed it ever was valid), and we hope this survey
will be adequate evidence of the successes of specialists in universal algebra. And
perhaps this survey will help dispel another (closely related) myth, which is
epitomized by Baer's remark [16, page 286], "The acid test for [a wide variety of
methods in universal algebra] will always be found in the theory of groups." Many
interesting results and ideas here either collapse completely or become hopelessly
complicated when applied to groups; but there is no lack of interesting classes of
algebras defined by equations to which the theories may be applied, as we shall see.
And this again is one of the attractions of the subject.
The writing of this survey was supported, in part, at various times, by the
University of Colorado, the Australian-American Educational Foundation and the
National Science Foundation. An early ancestor of this survey was a report of the
Australian S. R. I. lectures which appeared in the proceedings of the Szeged Universal
Algebra conference of 1975.
Although we believe the matehal unfolds rather naturally in the order we present
it, only õ õ 1,2, 3, 5 are essential to read first.