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

WALTER TAYLOR 49
For some more examples consult [429]; also cf. 14.1 above. In [429] there is a
fairly complete analysis of the influence of the laws obeyed by a topological algebra
on the laws which must be obeyed by its homotopy groups. wierczkowski's method
[407] of topologizing free algebras was essential to this work. But we are still a long
way from understanding the general interaction between the space of A and Eq A.
There is of course no a priori reason for believing that Eq A is especially important
here (rather than say the full first order theory of A), but experience has often yielded
examples which involved identities. (For e.g. connected fields obeying px = 0, see
[438], [320] .)
Very closely connected is the study of functional equations - see Aczl's book
[1] - a vast subject in itself. It proceeds like the above, also allowing certain
"constants," i.e. function symbols whose meaning is prescribed in advance, such as
ordinary addition + of real numbers - a typical early result being Cauchy's theorem
that the continuous solutions of
f(x + y) = fix) + f(y)
on the real numbers are the linear functions f(x) = ax.
17. Miscellaneous. Here we list a few topics that are concerned with equations
in one way or another, but do not fit precisely into any of the earlier sections.
17.1. Algebraically closed algebras are defined analogously to algebraically
closed fields. See e.g. Simmons [399], Bacsich [12], Forrest [139], Sabbagh [381]
and Schupp [386] and references given there. All algebraically closed algebras in a
variety V are simple iff every algebra in V can be embedded in a simple algebra of V.
(B. H. Neumann had earlier proved that every algebraically closed group is simple.)
(Cf. the final result of Evans in õ 12 - Word Problems, and that of McKenzie and
Shelah in 14.8.) Of course the satisfiability of equations (in algebraically closed fields)
is historically where the study of equations arose. In a recursively axiomatized variety
V, if a finitely presented A C V is embeddable in every algebraically closed B c V,
then A has solvable word problem (Macintyre [271]; see also [329], [381] and [386] t
17.2. Satisfiability of equations, especially their unique satisfiability, figures
heavily in the work of Sauer and Stone characterizing concrete endomorphism
monoids. (See [383] .)