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

32 EQUATIONAL LOGIC
conversely, every finite distributive lattice with unique atom can be represented in this
way (H. P. Gumm, unpublished). T. Evans informed the author that there exists a
finite commutative semigroup with A(Z) infinite.
13.3. For F = group theory, A(F) is modular, but very complicated, and
moreover can be given structure beyond the lattice-theoretic (see [330] and 17.9
below). At and near the top A(F) contains
ß
ß
x=y
(x rn= I, xy=yx)
xy = yx
with m > 1, ordered by divisibility - the Abelian part. It turned out to be difficult to
prove that IA(F)I = 2 0. This was established by Vaughan-Lee [434] who found 0
irredundant equations (as in õ11), and independently, by Ol'shanski¾ [337] who
found 0 "independent" subdirectly irreducible groups. (Cf. 9.18 above.) The variety
of 3-nilpotent groups has been completely described - see J6nsson [220] or
Remeslennikov [ 376].
13.4. For Z = semigroup theory, many results are known; consult the survey by
Evans [124] for more detailed information. Biryukov (1965) and later Evans [120]
first proved that [A(;)I = 2 0 (also see [198] for a nice infinite irredundant set of
semigroup laws). Dean and Evans [106] proved that x(yz)= (xy)z is finitely
meet-irreducible in A, i.e., that A(Z) has a f'mitely meet-irreducible least element.
Burris and Nelson [79] (and later Jeek [211 ] ) proved that A(Z) contains a copy of
Iloo, the lattice of partitions on an infinite set, and hence obeys no special lattice laws.
The fact that A(Z) is non-modular can be seen from this sublattice - due to Jezek -
isomorphic to the smallest non-modular lattice, NS:
xy = zw
xy = xz ß I
xyz = xzy
xy = yx
/ xyz = xzy = zxy