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

4 EQUATIONAL LOGIC
A are taken as values of the "variables" x C X appearing in o and r, i.e. the usual idea
for familiar equations like the associative law.) An equationally defined class of
algebras, alias a variety, is a class V for which there exists a set Z of equations with
V=mod23=(A: for alleC23, A e).
(Here "mod 23" abbreviates "the class of all models of 23.")
THEOREM. (Birkhoff 1935 [48] ). V is a variety if and only if V is closed under
formation of products, homomorphic images and subalgebras.
This enormously important result, in a style almost unheard of at its time,
effectively began "model theory." (See e.g. Tarski [41 1].) It can be considered the
ancestor of almost all research described in this survey. And yet in his history of
modern algebra [52], Birkhoff alludes to it in half a line only!
This theorem has been followed over the years by many others of a similar
format - sometimes called "preservation theorems" since Birkhoff's theorem (together
with compactness) has the corollary that if a sentence 0 is preserved under formation
of homomorphic images, subalgebras and products, then 0 is equivalent to a
conjunction of equations. For instance Keisler and Shelah proved that a class L of
structures is definable by a set of first order sentences iff L is closed under the
formation of isomorphic structures, ultraproducts and ultraroots. (Keisler proved this
assuming the G.C.H., and Shelah [396] without. See also [7].) For many other
preservation theorems, see e.g. [276]. More in keeping with the algebraic results of
this survey are the following three theorems.
THEOREM. ([228] ;see also [366]). V is definable by regular equations if and
only if V is closed under the formation of products, subalgebras, homomorphic images
and sup-algebras.
(An equation is regular if and only if exactly the same variables appear on both
sides. The sup-algebra of type (nt)tC T (unique within isomorphism) is the algebra
({ 0,1) ,Ft)tGT, where for each t,
Ft(a 1 ,...,ant) =
0 if al an t
1 otherwise.)
THEOREM. ([ 151 ], [55]; see also [395] ). V is definable by linear equations if