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

30 EQUATIONAL LOGIC
The theory Z 1 ^ 232 is not finitely based, for (we omit the proof), every basis must
contain equations essentially the same as
V::kW
B. J6nsson [223] found two finitely based equational theories of lattices whose meet
is not finitely based (also found by K. Baker - unpublished, but see [354]). Whether
there exist such theories of groups is unknown. There do exist such theories among
"modal logics" but not among "intermediate logics" (W. J. Blok thesis).
By the one-one correspondence between varieties and equational classes set up at
the beginning of õ5, we could equally well have described A as the lattice of all
varieties under reverse inclusion, and sometimes it is helpful to view A this way. (And
sometimes A is taken to be ordered by (non-reversed) inclusion of varieties - we will
not do this here.)
It is of interest to know what the lattices A look like. It has become clear that
they are very complicated, as we will see. Burris [74] and Je.ek [207] have proved
that if the type (nt)tC T has some n t > 2 or if n t > 1 for two values of t, then A
contains an infinite partition lattice, and hence obeys no special lattice laws at all.
Thus, it has proved fruitful to proceed by studying some (often simpler) special
sublattices of A, namely for fixed Z, the lattice A(Z) of all equational theories D-- Z.
(Equivalently, as above, the lattice of all subvarieties of Mod Z.) There is only one 23
with [A(Z)[ = 1, namely Z = {x = y}. Theories Z with [A(Z)[ = 2, i.e.
A(E):
are called equationally complete. Since every A is an algebraic closure system and
{ x =y } is finitely based, every theory has an equationally complete extension, and
ß
thus the top of A consists wholly of replicas of the above picture. An algebra A is
equationally complete iff Eq A is equationally complete. It has been determined that
there exist many equationally complete theories (and algebras), in two senses. First,
Kalicki proved [233] that in a type with one binary operation there exist 2 0 distinct
equationally complete theories (and the corresponding number has been evaluated for