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

46 EQUATIONAL LOGIC
precise definition). The number of properties known to be Malcev-definable has been
growing rapidly - see [420] for a summary of those known up to 1973 and [73] for a
partial updating; also see [21 ], [44]. They include the following:
IAI divides IBI whenever A C_ B G V, B finite;
V has no topological algebras with noncommutative homotopy (cf. õ 16);
no A G V is a union of two proper subalgebras [ 73 ];
V has no nontrivial finite algebras (cf. 14.1).
The first three of these hold for all groups. The last one has the distinction that there
is no way to recursively enumerate a Malcev condition for it, as was observed by J.
Malitz - see [420, page 383]. See [420], [333], or [26] for a necessary and sufficient
condition for a property of varieties to be Malcev-definable which easily entails all the
above examples except the second (and many more).
Permutability and modularity of congruences have been important from earliest
times in universal algebra (cf. 14.10 above and for recent examples, see [173] [428]
and recent work of J. Smith). But the condition which has been most important
recently is distributivity; this importance stems from J6nsson's theorem [221] that if
X is any subset of a congruence-distributive variety then
S.I. c HSP(X) c_ HSU(X)
(here S.I. denotes the class of all subdirectly irreducible algebras (õ4) and U(X) is the
class of all ultraproducts of families of members of X). (Also see Baker [ 18 ] .) In many
cases this yields a very good representation theory in the sense of õ4, e.g. if X is a
finite set of finite algebras, in which case U(X) = X. (For the finite case, see also
Quackenbush [373] for a somewhat simpler proof; a similar argument had earlier been
known to A. F. Pixley.) Among many uses of this result has been the investigation of
congruence lattices (see especially 13.2 and 13.11 above), and the "internal" model
theory of many individual varieties whose algebras have the operations of lattice
theory among their operations. See e.g. Davey [97], Berman [45], and references
given there. A very important kind of algebra generating a congruence-distributive
variety is a quasi-primal algebra, i.e., within equivalence, an algebra A = (A,T,F 1 ,F2,...)
T(x,y,z) =
where A is a finite set and { x zifx=y
if x % y