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