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

WALTER TAYLOR 43
for the failure of this fact for varieties of infinitary algebras.)
Some residually small varieties' Abelian groups, commutative rings with a law
x m = x (cf. õ 6), semilattices, distributive lattices, various "linear" varieties (as in 14.3
above); also if V = HSP A for A finite and V has distributive congruences (e.g. A = any
finite lattice), then V is residually small (by J6nsson's Theorem in õ 15 below). Also
any -product of two residually small varieties as in 9.14) is residually small.
(Similarly for AP and CEP.) Some non-residually small varieties: groups, rings,
pseudocomplemented semilattices (see [216], [ 382] ), modular lattices, and HSP A
for A either 8-element non-Abelian group (both generate the same variety - see 8.5
above). Also cf. 12.12 above.
A variety V is residually small iff every A in V can be embedded in an
equationally compact algebra B [419]. Mycielski [321] defined B to be equationally
compact iff every set [' of equations with constants from B is satisfiable in B if every
finite subset of [' is satisfiable in B. Here is an example [321] of failure of equational
compactness in the group of integers:
3x 0 + x 1 = 1
x 1 = 2x 2
x 2 = 2x 3
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One can solve any finite subset of these equations in integers simply by solving 3x 0 +
2n-lxn = 1, always possible; but clearly the entire set implies 3x 0 = 1, impossible.
Equational compactness is implied by topological compactness (use the finite
intersection property for solution sets), but not conversely. Thus we are led to this
problem, which has been settled positively for many V.
PROBLEM. [419]. If V is residually small, can every algebra in Vbe embedded
in a compact Hausdorff topological algebra? (See õ 16 below.)
There is a large body of research on equational compactness which we cannot
begin to cover here. See the survey review [423], or [72] or [425] for references.
Also see G. H. Wenzel's appendix to the new edition of [ 163 ].
Among the equationally compact B D_ A there is one which is "smallest," i.e., a
"compactification of A" - see [419, page 40] or [29]. Wglorz [439] proved that this
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