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

6 EQUATIONAL LOGIC
is closed under the formation of products and subalgebras.
Fisher has in fact shown [ 138] that we can always take this class of formulas to
be a setill Vopnkag principle holds. (This is one of the proposed "higher" axioms of
set theory.) Some less conclusive results about classes closed under the formation of
products and subalgebras occur in [196], [ 177], and [ 178]. Some mistakes of [ 155]
are corrected in [327].
For some other infinitary (in this case, topological) analogs of Birkhoff"s
theorem, see [ 112], [108], and [428]. (A unified treatment appears in [ 109] .) For
instance, the condition
(*) n ! x 0
defines a class of topological Abelian groups (here means "converges to"), which
contains all finite discrete groups but not the circle group. In [428] there is a theory
of classes defined by conditions similar to (*); these classes are called "varieties" of
topological algebras.
algebra.
See [57] for another analog of Birkhoff's theorem which goes beyond pure
4. Generation of varieties and subdirect representation. Let V 0 be any
collection of algebras of the same type. Since the intersection of any family of
varieties is again a variety, there exists a smallest variety I,' D_ V 0. It clearly follows
from õ3 that
I,' = Mod Eq V 0,
where Eq V 0 means the set of equations holding identically in V 0. It is also very easy
to prove (using Birkhoff's theorem of õ 3) that
V = HSP
since, as one easily checks, the R.H.S. is H-, S- and P-closed. Problem 31 of Gnitzer's
book [ 163] asks whether this fact implies the axiom of choice. (For M any class of
algebras, HM, SM, PM denote the classes of algebras isomorphic to homomorphic
images, subalgebras and products of algebras in M.) In practice, S and P seem natural
enough, affording a "co6rdinate" representation of [some] algebras in V using
algebras of V 0. But H seems less natural, and one hopes in favourable circumstances to
avoid it, arriving at