Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
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WALTER TAYLOR 19<br />
õ7, this easy result corresponds to the fact that the product <strong>of</strong> two finitely presented<br />
groups is finitely presented.<br />
9.15. Recently Murskii has proved [319] that "almost all" finite algebras have a<br />
finite base for their identities (i.e., for fixed type, the fraction <strong>of</strong> such algebras among<br />
all algebras <strong>of</strong> power k approaches 1 as k - oo _ or even, for fixed k, as the number <strong>of</strong><br />
operations approaches oo). (The unary case is easy - all are finitely based; in the<br />
non-unary case he in fact proves much more: almost all are quasi-primal - cf. õ15<br />
below, and also 9.11 and 10.7.)<br />
For some further remarks about finite algebras with finite bases, consult [235].<br />
We now turn to equational theories which are not finitely based. Of course it is almost<br />
trivial to construct such theories using infinitely many operations Ft(t 6 T), even some<br />
which are equivalent to the (finitely based!) theory with no operations. As G.<br />
Bergman pointed out,non-finitely based theories with finite T arise almost automatically<br />
if we consider a semigroup S which is finitely generated (say by F C_S), but not<br />
finitely related. Our theory can be taken to have unary operations for f F and laws<br />
flf2 "' fk x= fk+l "' fs x whenever fl '" fk = fk+l "' fs in S. Some more<br />
interesting research has centered on finding less obvious, but more important,<br />
examples <strong>of</strong> theories and algebras which have T finite and are still not finitely based:<br />
9.16. The algebra with universe (0,1,2) and binary operation:<br />
(Murski]' [ 318 ]<br />
9.17. The<br />
o I ;<br />
0 0 0 0<br />
I 0 0 I<br />
, following Lyndon [266] ).<br />
six-element semigroup<br />
0<br />
(with ordinary matrix multiplication). (Perkins [349].) Earlier Austin [ 11 ] gave some<br />
other varieties <strong>of</strong> semigroups which are not finitely based. C. C. Edmunds has recently<br />
shown that six is as small as possible for a semigroup with zero and unit.