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

WALTER TAYLOR 13
(2,1) is obviously not intrinsic to the idea of a group. Many of the properties
considered below are (obviously) equivalence-invariant, but a few, such as being
"one-based" (õ10) are not, as we shall see, comparing 10.3 and 10.10 below.
Moreover, certain results cannot even be stated without mentioning equivalence (14.2
and 14.5 below).
It is possible to define equational classes so as to make all expressable properties
automatically equivalence-invariant, i.e. to give no preference to any of the possible
equivalent forms of a given variety. This amounts to considering the set (rather than a
sequence (Ft)tCT) of all possible operations defined by V-terms. This idea goes back
to P. Hall (see [91, pages 126-132] ), and has been worked out independently in detail
by W. D. Neumann [332] and F. W. Lawvere [254] (see page 362 of [420] for more
detailed historical remarks, and pages 390-392 for a proof- independently found by
W. Felscher - of the equivalence of these two approaches). Some of these ideas were
presented independently by Claude Chevalley in a speech at Stanford in November,
1962. Certainly Lawvere's approach came much sooner than Neumann's and has
obtained a much wider following. We will not describe his invention, "algebraic
theories," except to say that they contain precisely the right amount of information
to describe varieties without allowing any individual operations to play a special role.
For further references see [420, loc. cit.]; see also [272].
Despite some enthusiastic claims (see e.g. the dustjacket or Chapter 3 of [346] or
[443, page 121]) that these category-theoretic ideas would take over the study of
universal algebra, this hasn't really happened by 1979. Their significant role, so far,
has been to suggest analogies outside pure algebra (e.g. compact Hausdorff spaces).
But they have had almost no impact yet in the study of ordinary varieties (i.e. the
kinds of subjects discussed in this survey), with one interesting exception: the study of
Malcev conditions (õ 15 below) was facilitated by viewing it as a study of morphisms
between algebraic theories (see [420] and [427] ). Other possible directions are given in
[347], [40], [58], and [244]. Some useful remarks are found in Lawvere [255]. The
reason that direct application of "algebraic theories" to equational logic is difficult (or
unnecessary) lies mostly in its model theory; to see this, let us notice (as many others
have before) that the passage