COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

268 Donald E. Knuth and Peter B. Bendix Word problems in universal algebras 269
are involved, we can have unrelated words such as
.&wwlv1v2 # f32.Z2'2Vl, (2.4)
f2w'z # f2v2v1, (2.5)
where f& fi fs are operators of degrees 2, 3, 5 respectively.
The principal motivation for the given definition of a z B is the following
fact:
THEOREM 2. If a > B then s(el, e2, . . . , en; g) > s(el, e2, _ ..,e,; /%for
all words f&, . . ., 0,.
Proof Let U’ = s(el, e2, . . . , 8,; E> and /I’ = s(el, e2, . . . , 8,; p).
If condition (1) holds for x and B, then it must hold also for a’ and p’.
For in the first place, every word has weight Z= We, so
w(4 = w(x) fjzl n(vj, sc>(+@j> - ~0)
=- w(B) + j~ln(V.i, B)(w@j> - wo) = WW.
Secondly, fl(Vi, 4 =j;l n(vj> x>n(vi, ej> aJzln(vj2 &(Vi, ej> = n(Vi, B’>.
If condition (2) holds for M. and ,!I, then similarly we find ~(a’) = w(/?‘)
and n(vi, a’) = n(~i, /3’) for all i, and CC’ =&I . . . a& b’ = fk& . . . &
where CC: = S(&, . . ., 6,; a,) and p: = S(&, . . ., 6,; brs,> for all r. Hence
either j =- k, or an inductive argument based on the length of a will
complete the proof.
Corollary. There is no infinite sequence of words such that a, r a2 >
=-as> . . . . For if there were such a sequence, we could substitute a
nullary operator f for each variable vj, j 3 1; Theorem 2 implies that this
would give an infinite descending sequence of pure words, contradicting
Theorem 1.
It should be emphasized that Theorem 2 is a key result in the method
which will be explained in detail in subsequent sections; and the fact that
cz # fl can occur for certain words a and /3 is a serious restriction on the
present applicability of the method. The authors believe that further theory
can be developed to lift these restrictions, but such research will have to
be left for later investigations.
It may seem curious that f5 ~1 vlvlvlvz # f3v2v2vl; surely f5vlvlvlvlv2
appears to be a much “bigger” word than f3v2v2v1. But if we substitute a
short formula for VI and a long formula for v2, we will find f&v2v1 is
actually longer than f5v1v1vlvlv2.
Theorem 2 is not quite “best possible”; there are words a and /I for
which a # B yet S(&, e2, . . . , 0,; a) z S(&, e2, . . . , 0,; @) for all “pure”
words el, . . . , en. For example, consider
f3Vl # fifl (2.6)
where f3 and fi are unary operators of weight one, and fl is a nullary
operator of weight one. If we substitute for v1 a pure word 8 of weight 1,
we have f30 =-JZfi by case (2a); but if we substitute for v1 any word 6
of weight greater than one, we get f& =- fifi by case (1). We could therefore
have made the methods of this paper slightly more powerful if we had
been able to define fgrl =- fifi; but such an effort to make Theorem 2
“best possible” appears to lead to such a complicated definition of the relation
0: P- p that the comparatively simple definition given here is preferable.
So far in practice no situation such as (2.6) has occurred.
Let CI and ,4 be words with V(E) < n, v(p) G n. In the following
discussion we will be interested in the general solution of the equation
sh . . . . 8,; 4 = s(el, . . ., 0,; b) (2.7)
in words or, . . ., 0,. Such an equation can always be treated in a reasonably
simple manner:
THEOREM 3. Either (2.7) has no solution, or there is a number k, 0 G k s n,
and words 01, . . . , a, with ~‘(0~) =S k for 1 4 j == n, where
{R, g2, . - ., Vk} c (01, . . ., G},
(2.8)
such that all solutions of (2.7) have the form
6j==S(vr ,..., pk;Uj), lGj