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

48 EQUATIONAL LOGIC
16. Connections with topology. If A = (A,Ft)tC T is any algebra and T is any
topology on A such that every F t' A nt-* A is continuous (in the nt-fold topological
product of T), then we say that A = (A,T,Ft)tC T is a topological algebra. The space
(A,T) is not at all independent from the equational theory of A, but rather the two
seem to influence each other quite strongly. This influence is poorly understood, but
many interesting examples are known.
16.1. No sphere except S 0, S 1 , S 3, S 7 can be an "H-space," i.e., can obey
ex = x = xe
for binary multiplication and constant e (Adams [2]), and S 7 cannot also obey the
associative law (James [204] ).
16.2. The space of a topological group must be homogeneous, with Abelian
fundamental group, and if compact and uncountable, of power > 2 q0. (All of these
facts are essentially well known.)
16.3. The space of a topological Boolean algebra, if compact, must be a power
2 n for 2 a 2-element space. (Kaplansky [237] .)
16.4. The space of topological semilattice has zero homotopy in each of its
components in each dimension (Taylor [429]), and if compact, connected and
finite-dimensional, cannot be homogeneous (Lawson and Madison [253] ).
16.5. If V is defined by the equations (*) of 14.3 above, then it is not hard to
check that the topological algebras in V have "square" universe (as in Evans [ 118] ),
i.e. each is homeomorphic to the square of some other space.
16.6. If V is the product U W of two varieties (see [424], õ0]), then a
topological algebra in V with product-indecomposable space must be either in U or W
(this can be seen fairly easily from the methods outlined in [420, pages 357-358] or
[424, pages 265-267] ).
16.7. If a compact connected topological algebra obeys the law
(xy)(yz) = xz,
then it also obeys the law xy = uv. (Bednarek and Wallace [38] .) It is an old problem
of A.D. Wallace (see [378] ) whether the "skew-associative" law
x(yz) = (zx)y
can hold on the unit interval without the associative law holding as well.