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

WALTER TAYLOR 27
(Typically 0 is of the form E 1 U 2, where 1 is a set of laws not involving (Ci)iC I
and 2 is a set of equations with no variables, viewed as "relations" on the
"generators" Ci(i C I). One then speaks of "the word problem for [this presentation
of] the algebra FEi(Ci)/(22)", where (2) means the smallest congruence containing
all pairs of terms in E2.) Post [367] and Markov [285] proved that there exists a
semigroup with undecidable word problem (i.e. that one may take E 1 to be the
associative law). Much more difficult was the 1955 result of Boone and Novikov [61],
[334] (see also [65] and [302]) that there exists a group with unsolvable word
problem. Notice that all the results on (un)decidability of equational theories
mentioned above are really a special kind of word problem result (with III= 0' 2 =
G. Hutchinson [J. Algebra , 26(1973), 385-399] and independently L. Lipshitz
[Trans. Amer. Math. Soc., 193(1974), 171-180] have shown the existence of a finitely
presented modular lattice with unsolvable word problem. Hutchinson later gave an
example with five generators and one relation [Alg. Univ., 7(1977), 47-84]. As we
mentioned above, R. Freese subsequently reduced th, number of relations to zero.
Evans [116],[117] proved that the word problem is solvable for E 1 (i.e.,
uniformly for all E 2) iff it is decidable whether a finite partial algebra obeying the
laws of E 1 (insofar as they can be evaluated) can be embedded in a full algebra
obeying
1 (see also [ 163, õ30]). Thus lattices have solvable word problem. Clearly
the condition holds if finitely generated algebras in ! are always finite, or if finitely
presented algebras are always residually finite (i.e. embeddable in a product of finite
algebras), and for this last case there is a "local" version: the word problem is solvable
for A = Fi (Ci)/( 2) with {C i , 2 finite, if A is residually finite (Evans [ 121 ] ). For
applications see e.g. [ 148], [260].
Following work of Boone and Higman in group theory [63], Evans [127]
recently proved that an algebra A has solvable word problem iff A can be embedded in
a finitely generated simple algebra B which is recursively presented. (Generally we
cannot demand that B HSP A or B V for V any preassigned variety containing A.)
(Cf. 14.8 and 17.1 below.) Also see [129].
PROBLEMS ON FINITE ALGEBRAS. Kalicki proved [232] that it is decidable,