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

WALTER TAYLOR 23
beginning of õ7) (Green and Tarski [ 172], [413] ). (Cf. 10.3 above.)
10.11. A: ((0,1} ,v ) and A = ((0,1} ?) with
(Potts [368] ).
ß - 0 I
0 I I
I 0 I
10.12. If 23 is a finitely based theory of type (ml,m2) with ml,m 2 2 in which
F 1 and F 2 are each idempotent, i.e.
23 [- Fi(x,x,...,x) -- x (i = 1,2),
then 23 is 2-based (and may also be 1-based) (Padmanabhan [340] ).
10.13. If 23 is a finitely based theory with a majority polynomial (as in 10.2
above), then 23 is 2-based (and sometimes 1-based) (Padmanabhan and Quackenbush
[342]). (McKenzie [291] had this result for varieties in which lattices are definable.)
Isolated results: Lattices are definable in 2 equations using only 3 variables
(Padmanabhan [339] ). Two variables will not suffice for lattices (see [ 165, page 62] ),
nor for Boolean algebras (Diamond and McKinsey [ 107] ). Cf. also [372] and 14.4. If
230 is
x(yz) = (xy)z
XX=X
then every theory _ 23 0 has a basis consisting of 23 0 U { o }, i.e. 23 0 together with one
more axiom (Biryukov [541, Fennemore [137], and Gerhard [152]). (Cf. 13.6
below. )
PROBLEM. (Specht). Does there exist a non-finitely based variety of rings?
1 1. Irredundant bases. 23 0 is an irredundant base for 23 iff 23 0 is a base for 23 but
no proper subset of 23 0 is a base. Tarski [413] has defined
V(23) = { [230l: 23 0 is an irredundant base of 23}.
(Here [I denotes cardinality.) Tarski's interpolation theorem [413], [414] states that
V(23) is always an interval (see [3091 for a connection between this and some other
interpolation theorems, especially in graph theory; see also [1 57] ). One easily checks
that (at least for a type (nt)tC T with T finite), either (23) = 0, x7(23) = {