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

28 EQUATIONAL LOGIC
for arbitrary finite algebras A, B of finite type whether
HSP A = HSP B.
And Scott proved [390] that it is decidable whether finite A is equationally complete.
Almost all interesting decision problems about finite A are open, for instance:
PROBLEM 2. (Tarski). Is it decidable whether a finite algebra A of finite type
has a finite base for its equations? (Cf. 89.) (And see Perkins [348] .)
whether
or whether
PROBLEM 3. (Pixley, et al.). For a finite A of finite type, is it decidable
HSP A = SPHS A,
HSP A = SP A?
(For the relevance of this question, consult õ 84,15.)
PROBLEMS ON FINITE SETS OF EQUATIONS. Many undecidability results
are known on finite sets ; of equations; for a full report we must refer to the chart in
the introduction of McNulty [306]. Here we list a few undecidable properties of .
12.1. ;« x=y(Perkins[348]).
12.2. ; has a finite non-trivial model (McKenzie [296] ).
12.3. There exists finite A with ; a base for Eq{A) (Perkins [348] ).
12.4. ; is equationally complete (cf. 813 below)(Perkins [348] ).
12.5. ; is one-based (cf. 810) (Smith [403], McNulty [306]).
12.6. The equational theory deduced from ; (i.e., Eq Mod 2 = {e: ;«e}) is
decidable. (Perkins [348 ] .)
12.7. Mod 2 has the amalgamation property (defined as in group theory -see
14.6 below) (Pigozzi [356] ).
12.8. ; has the "Schreier property," i.e., subalgebras of free algebras are free
(see 14.12 below). (Pigozzi [356].)
12.9. (For certain fixed theories F, e.g., F = group theory) ; is a base for F.
(McNulty [306] MurskiI [316]). (But it is decidable whether ; is a base for xy =yx
(Ng, Tarski - see [413] ).
12.10. Mod Z has distributive congruences (cf. 815 below) (McNulty [306 ] ).
12.11. Mod ; is residually finite (McNulty [306] ).