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

40 EQUATIONAL LOGIC
semilattices (co n = 2n-l), the variety of groups given by the law x 3= 1 (see e.g.
[330] ), the variety of Heyting algebras defined by "Stone's identity" (see [ 189] ), and
some varieties of interior algebras (W. J. Blok, thesis). The quasi-primal varieties of
Pixley et al. often have very easily calculated co (see e.g. [362] or [369] for
quasi-primal varieties); see the chart on page 291 of [424] for some explicit
calculations. (But in the eight line, 1 should be 0.)
But for most common varieties, the invariant co(V) is either trivial (because
infinite) or hopelessly complicated. Sometimes special cases can be calculated.
Dedekind found in 1900 that the free modular lattice on 3 generators has 28 elements
("free algebra" had not yet been defined) (see [50, page 631 ). For some other special
calculations (distributive lattices, etc.), see [50, page 63], [437], [46] and [47].
The class of all finite co(V) is closed under (co6rdinatewise) multiplication (see e.g.
[424, 0.5(4), page 266]), and it forms a closed set in the space coco (S'wierczkowski -
see [ 282, page 181 ] ).
The famous Burnside Problem asked whether F v(n) is always finite for V the
variety of groups defined by the law x TM = 1. The negative solution by Novikov and
Adjan [335] stated that IFv(2)I-- q0 when e.g. m = 4381. The proof in [66] is said
to be false (see [3] ). (Now 4381 has been reduced to 665 [3] .) For related results in
semigroups, see [ 171 ].
In general algebra, a typical theorem is that of Ptonka [364]: if con(V) = n. 2 n-1 ,
then V must be equivalent to one of four varieties, namely those given by 231 - 234:
Zi: xx=x
(xy)z = x(zy)
x(yz) = x(zy)
Z2: xx = x
(xy)z = (xz)y
x(yz) = xy
(xy)y = xy
233: xx = x
(xy)z = (xz)y
x(yz) = xy
(xy)y = x