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

WALTER TAYLOR 9
derivation system for "equational logic" which has been useful for proving many of
the undecidability results of {}12. For semigroups it was originated by Kuro (see
[251]); see also [3052, [306]. We first assume 2 is closed under forming
substitutions. By a "derivation" we mean a sequence (o 1,...,on) such that for i =
1,2,...,n-l, there exists (o =/3) C 2 such that o (or/3) is a subterm of o i and oi+ 1 results
from o i by replacing the subterm a by /3 (resp. a). Then if we define 21- o = r to
mean that there exists a derivation (o ! ,...,On) with o I = o and o n = r, then the above
theorem of Birkhoff remains true.
Proof calculi have also been useful in establishing some interpolation and
definability results [1422, [200], [187]. Some (more computational) rules are given
in Knuth and Bendix [248] and used in Glennie [158]. See also [43, {} 102.
Theorems parallel to this completeness theorem of Birkhoff are not numerous.
There is of course G6del's complete set of rules of proof for first order logic. (For
second order logic, no such set of rules can exist, but see Karp [2382, Keisler [242]
for some rules which go beyond first order logic.) A. Selman [393] has given a set of
rules for equation implications (independently discovered by D. Kelly [unpublished] ),
and Sfomirski gave an infinitary analog of Birkhoff's theorem in [402]. G. McNulty
has asked [3072 whether a simple set of rules exists for the class of all positive
sentences. See any logic book (e.g. [267] or [85] ) for more information on I-. A large
portion of this survey is concerned with the relation I-; in the next section we will
specifically illustrate Birkhoff's completeness theorem.
6. Examples of . Let P denote the axioms of commutative associative ring
theory. We will show that
(*) P,x 48=x x 2=x.
(Definition in {}5.) For this it is enough to prove that every ring obeying x 48 = x also
obeys x 2 = x, and by Corollary 2 in {}4, it is enough to check subdirectly irreducible
rings obeying x 48 = x. Such a ring clearly has no non-zero nilpotent elements, and so
by the exercise in {}4, must be a field, having q < 48 elements. One easily checks that
for some m, m(q - 1) + 1 = 48, i.e. m(q - 1) = 47 and so either (q - 1) = 47, i.e. q = 48,
impossible since q must be a prime power; or q-1 = 1, i.e. q= 2, yielding the
two-element field, which does obey the law x 2 = x, proving (*). And so by Birkhoff's