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

44 EQUATIONAL LOGIC
compactification is always in HSP A.
14.9. The conjunction of AP, CEP and residual smallness (14.6 - 14.8) is
equivalent to the purely category-theoretic property of injectire completeness (see
Banaschewski [30] ). (Pierce [353] noticed that injective completeness implies AP.) V
is injectively complete (or, "has enough injectives") iff every algebra in V is
embeddable in a V-injective, i.e. an algebra A C V such that whenever B C_ C C V and
f: B A is a homomorphism, there exists an extension of f to g: C A. (See e.g.
[ 143, õ6.2] but remember that most varieties are not Abelian categories.) The variety
of semilattices has enough injectives (Bruns and Lakser [68], Horn and Kimura
[190] ) and so does that of distributive lattices (Banaschewski and Bruns [32], Balbes
[22]). For a theory of injective hulls in varieties, see [418,page 411 ]. See also [154].
We know examples to show that AP, CEP and residual smallness are completely
independent properties, except for one case'
PROBLEM. AP and Res. Small CEP?
For further information on varieties with enough injectives, see Day [99] and
Garcih [150]. In 14.10 just below we will mention another category-theoretic
property of varieties. Yet another one is that of being a binding category, investigated
for varieties in [398] and [ 180].
14.10. Unique factorization of finite algebras (UFF) in V, i.e., if A 1 X -.. X
A n- B 1 X --. X B s V is finite and no A i or Bj can be further decomposed as a
product of smaller factors, then n = s and after suitably tenumbering, A 1 - B1, A 2
B 2 ..... A n - B n. (Historically, this problem has been approached independently of any
mention of V, but the results obtained often have an equational character.) Birkhoff
proved [ 50, page 169] that V has UFF if V has a constant term a such that
(*) V F(a ..... a) = a for all operations F of V
and V has permutable congruences, and Jdnsson [222] improved "permutable" to
"modular" (see õ15 for these terms). Jdnsson and Tarski [230] proved that V has
UFF if V has a constant term a obeying (*) and a binary operation + such that
Vx+a=x=a+x.
McKenzie showed [294] that the variety of idempotent semigroups (see 13.6) has
UFF, but the variety of commutative semigroups does not (see [50, page 170]). UFF