Some open questions in algebraic logic
Attributions are to the best of my knowledge  corrections welcome, as
are solutions.
Below, unless otherwise stated, n denotes a finite integer.

Is RRA axiomatisable in firstorder logic with n variables, for any
finite
n?

Is SRaCA_{5} closed under completions?
(For n>5, SRaCA_{n }is not closed under completions.)
 Is RA_{5} closed under completions?
(For n>5, RA_{n }is not closed under completions.)
 Let n>4. Is there a set of canonical equations
axiomatising SRaCA_{n}? Or RA_{n}?
 Is there a variety of BAOs that is closed under completions
(hence canonical) but with no canonical axiomatisation?
 Str RCA_{n} is the class of atom structures
of full complex algebras that are representable ndimensional cylindric algebras.
For infinite n, is Str RCA_{n} an elementary class?
(For n<3, Str RCA_{n} is elementary,
and for finite n at least 3, Str RCA_{n} is nonelementary.)

It is known that RA_{n }properly contains SRaCA_{n}
for
each n>4, and that SRaCA_{n }is not finitely axiomatisable
over
RA_{n}, but that the intersection of all RA_{n} is RRA,
which is contained in any given SRaCA_{n}.
Is there a function f:\omega > \omega such that RA_{f(n)}\subseteq
SRaCA_{n} for all n>4?
Is there a recursive f?
 Is every algebra in SRaCA_{n} (n>4) embeddable in an atomic relation algebra
with a ndimensional
cylindric basis?
(Not every atomic algebra in SRaCA_{n} has a ndimensional
cylindric basis itself. E.g., a representable projectiveplane Lyndon
algebra
with at least 6 atoms has no 5dimensional cylindric basis. But it
clearly
embeds in an atomic relation algebra with such a basis.)
 [R Maddux] Is it decidable whether a finite relation algebra has
a finite
representation (i.e., with finite base)?

For fixed finite n at least 5, is it decidable whether a finite
relation
algebra is a subalgebra of the relation algebra reduct of a finite ndimensional cylindric
algebra?
(It is undecidable whether a finite relation algebra is a subalgebra of the relation algebra reduct
of
a possibly infinite ndimensional cylindric algebra  i.e., is in SRaCA_{n}.)
 [R Maddux] What is the limiting probability of a finite
relation algebra being representable?
Problems of T S Ahmed:

Is there an uncountable atomic algebra A in Nr_{3}CA_{\omega}
which has no complete representation? (For countable algebras the
answer is `no'.)

Is the intersection of Nr_{3}CA_{n} for finite n equal
to Nr_{3}CA_{\omega}?

What can be said about the number of different CA_{\omega}s
that
a given CA_{3} can be a neat reduct of?
Ian Hodkinson, Nov 2008, amended Mar 2012