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

22 EQUATIONAL LOGIC
some algebras or theories which are one-based:
10.1. The variety of all lattices (McKenzie [291]). McKenzie's original proof
yields a single equation of length about 300,000 with 34 variables. Padmanabhan
[341 ] has reduced it to a length of about 300, with 7 variables. Here we mean lattices
formulated as usual with meet and join. Cf. 10.8 and 10.9 below. More generally:
10.2. Any variety which has a polynomial m obeying
m(x,x,y) = m(x,y,x) = m(y,x,x) = x
(a "majority polynomial") and is defined by "absorbtion identities," i.e., equations of
the form x = p(x,y,...). (McKenzie [291 ]; see also [341 ].)
10.3. Any finitely based variety V of EEl-groups (see the beginning of õ7)
(Higman and Neumann [ 185] ). Tarski got this for V = all Abelian groups (see [413] ).
(Cf. 10.10 below.) For a recent proof, see [236].
10.4. Certain varieties of rings (with operators) (Tarski [413]). For some more
general formulations of 10.3 and 10.4, see Tarski [413 ].
10.5. Boolean algebras. (Grgtzer, McKenzie and Tarski) (see [165, page 63]).
(Cf. [401 ].) (Also cf. 10.6 and 10.7.)
10.6. Any two-element binary algebra except (within isomorphism) as in 10.11
below. (Potts [368] .)
10.7. Every finitely based variety with permutable and distributive congruences
(McKenzie [296]; Padmanabhan and Quackenbush [342]). By 9.11 this applies to
any finite algebra which generates a variety with permutable and distributive
congruences, e.g. a quasi-primal algebra (see [362], [369]). Primal algebras were
already known to Gritzer and McKenzie [168]. ([296] contains some very interesting
special one-based varieties.)
Here are some theories (and algebras) which are 2-based but not I-based:
10.8. The variety of all lattices given in terms of the single quaternary operation
Dxyzw = (xv y) ^ (z vw) (McKenzie [291]). (Cf. 10.1.)
10.9. Any finitely based variety of lattices other than the variety of all lattices
and the trivial variety defined by x =y (McKenzie [291]). (Here again we mean the
usual lattice operations.)
10.10. Any non-trivial finitely based variety of I'2-groups (defined at the