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

2 EQUATIONAL LOGIC
2. The existence of free algebras. Let V be any class of algebras of fixed type.
By definition a V-free algebra on the set X (denoted Fv(X)) is an algebra B =
(B,Ft)t- T such that
(1) B c V;
(2) X c_ B;
(3) if A c V and 0: X-+A is any function, then there exists a unique
homomorphism : B -+ A with ] X = 0'
One easily checks that if B and C are each V-free on X, then there exists a unique
isomorphism : B-+ C with the identity on X. (Thus we may say "the" V-free
algebra on X.)
THEOREM. [48]. If V is any non-trivial class of algebras closed under the
formation of subalgebras and products, then Fv(X) exists for every X.
A proof can be found in any of our references on general algebra. Birkhoff's
original proof has been abstracted in category theory to yield the "adjoint functor
theorem." See e.g., [ 143, page 84]. For some other adjointness results on classes V see
e.g., [432] and [346, pages 148-149]. Such generalized free objects (e.g. tensor
algebras and universal enveloping algebras) were important in Lawvere's development
of an invariant approach to this subject (see õ 7 below).
Familiar examples of free algebras are free groups and free AbelJan groups. Thus
in some (but not all) cases, elements of free algebras can be written as "words." (We
will return to this point in õ 12 below.) This more concrete description of free algebras
has also caught the attention of category theorists; see e.g. Gray [ 170].
It is unclear whether there is a successful generalization of the Theorem to more
general classes V. Gratzer [ 163, Chapter 8] proposed such a generalization for classes
V defined by a set 22 of first order sentences. But unfortunately his "free algebra" was
not independent of the choice of axiomatization [Colorado Logic Seminar, Spring,
1969, unpublished].
Perhaps the most remarkable recent result on free algebras is Shelah's [397]: if 3,
is a singular cardinal, every subalgebra of A with < 3, generators is free, and A has
cardinality 3,, then A is a free algebra.
Notational convention henceforth: K is the class of all algebras of the fixed type