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

2. xy = yx
3. x + (y+z)-- (x+y) + z
4. x(yz) = (xy)x
5. x(y+z) = xy +xz
6. x y+z = xYx z
7. (xy) z= xZy z
8. (xy)Z = x(yZ)
9. x'l =x
10. xl=x
11. lX=l.
WALTER TAYLOR 17
(Tarski - see [286] .) Tarski has called this the "high school identity problem (with
unit)." (Incidentally, Martin [286] has remarked that it follows easily from the
methods of Richardson [377] that
Eq(oo, 1 ,x+y,xy,xY) = Eq(R +, 1 ,x+y,xy,xY),
where R + is the set of non-negative real numbers.)
We could go on and on with interesting examples (see e.g. [4], [104], [135],
[ 153], [ 188], [265], and [408] ), but we will stop here. In place of finding a base
of a given A, one can often be content with the knowledge that a finite 0 exists or
does not exist, as the case may be: this is the idea of the next section. (Although
sometimes explicit - but complicated - bases are found in the researches reported in õ 9
and õ10. The methods of õ 14.5 and õ15 also sometimes lead to finding 20 and A as
in this section.) The reverse problem, of finding a generic A for a given 20' is less well
defined. As remarked above, A always exists, but finding a "known" (i.e. familiar) or
simple generic algebra can be very elusive, e.g. for modular lattices. (The problem of
"simply" describing a free algebra is really a word problem - see õ 12 below - and the
word prcblem for free modular lattices is not solvable.)
9. FinRely based theories. We say that an equational theory 2 is nitely based
iff there exists a finite set 20 of axioms for 2. (The definitions in õ7 should make it
clear that this is an equivalence-invariant property of all 2 which have finitely many
operations.) Evidently many familiar theories are finitely based - groups, Boolean
algebras, rings, lattices, etc.; see also the various examples in õ8. Here we list some