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

WALTER TAYLOR 5
and only if V is closed under the formation of products, subalgebras, homomorphic
images and co mp lex algebras.
(An equation is linear iff each side has at most one occurrence of every variable.
If A = (A,Ft)tC T is any algebra, the complex algebra of A is B = (B,Gt)tC T, where B is
the set of non-emp ty subsets of A, and
Gt(u 1,...,unt) = { Ft(a 1 .... ,ant): a i u i (1
i nt) .)
The next important result really goes back to J.C.C. McKinsey [303] in 1943 (he
proved a theorem which, in combination with the above theorem of Keisler and
Shelah, immediately yields our statement). The present formulation was probably first
given by A. I. Malcev [280, page 214], [279, page 29] ;many other proofs have been
independently given [166], [307], [394], [145], [28] - although the precise
formulation differs from author to author. See also [85, Theorem 6.2.8, page 337],
and for related results[33],[239],[186] and [187].
THEOREM. V is definable by equational implications iff V is closed under the
formation of products, subalgebras and direct limits.
An equational implication is a formula of the form
(e l&e 2&-..&en)-e,
where e,e 1,...,e n are equations, for example the formula
(xy = xz - y = z)
defining left-cancellative semigroups among all semigroups. For direct limits see
[163], [143] (or any other book on category theory). For some interesting classes
defined by equational implication, see [417] and [40]. For some infinitary analogs of
Birkhoff's theorem see [402], and of McKinsey's theorem, see [186]. In the next
result, infinitary formulas are in a sense forced upon one, even though it is a result
about ordinary finitary algebras. A generalized equational implication is a formula
ii ei - e,
where e, e i (i I) are equations (possibly infinitely many). The next theorem was
perhaps first stated in [33], although maybe some other people knew of it.
THEOREM. V is definable by a class of generalized equational implications iff V