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

WALTER TAYLOR 19
õ7, this easy result corresponds to the fact that the product of two finitely presented
groups is finitely presented.
9.15. Recently Murskii has proved [319] that "almost all" finite algebras have a
finite base for their identities (i.e., for fixed type, the fraction of such algebras among
all algebras of power k approaches 1 as k - oo _ or even, for fixed k, as the number of
operations approaches oo). (The unary case is easy - all are finitely based; in the
non-unary case he in fact proves much more: almost all are quasi-primal - cf. õ15
below, and also 9.11 and 10.7.)
For some further remarks about finite algebras with finite bases, consult [235].
We now turn to equational theories which are not finitely based. Of course it is almost
trivial to construct such theories using infinitely many operations Ft(t 6 T), even some
which are equivalent to the (finitely based!) theory with no operations. As G.
Bergman pointed out,non-finitely based theories with finite T arise almost automatically
if we consider a semigroup S which is finitely generated (say by F C_S), but not
finitely related. Our theory can be taken to have unary operations for f F and laws
flf2 "' fk x= fk+l "' fs x whenever fl '" fk = fk+l "' fs in S. Some more
interesting research has centered on finding less obvious, but more important,
examples of theories and algebras which have T finite and are still not finitely based:
9.16. The algebra with universe (0,1,2) and binary operation:
(Murski]' [ 318 ]
9.17. The
o I ;
0 0 0 0
I 0 0 I
, following Lyndon [266] ).
six-element semigroup
0
(with ordinary matrix multiplication). (Perkins [349].) Earlier Austin [ 11 ] gave some
other varieties of semigroups which are not finitely based. C. C. Edmunds has recently
shown that six is as small as possible for a semigroup with zero and unit.