# 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.

1. Is RRA axiomatisable in first-order logic with n variables, for any finite n?
2. Is SRaCA5 closed under completions?
3. (For n>5, SRaCAn is not closed under completions.)
4. Is RA5 closed under completions?
5. (For n>5, RAn is not closed under completions.)
6. Let n>4.  Is there a set of canonical equations axiomatising SRaCAn?  Or RAn?
7. Is there a variety of BAOs that is closed under completions (hence canonical) but with no canonical axiomatisation?
8. Str RCAn is the class of atom structures of full complex algebras that are representable n-dimensional cylindric algebras.  For infinite n, is Str RCAn an elementary class?
(For n<3, Str RCAn is elementary, and for finite n at least 3, Str RCAn is non-elementary.)
9. It is known that RAn properly contains SRaCAn for each n>4, and that SRaCAn is not finitely axiomatisable over RAn, but that the intersection of all RAn is RRA, which is contained in any given SRaCAn.
10. Is there a function f:\omega -> \omega such that RAf(n)\subseteq SRaCAn for all n>4?
Is there a recursive f?
11. Is every algebra in SRaCAn (n>4) embeddable in an atomic relation algebra with a n-dimensional cylindric basis?
(Not every atomic algebra in SRaCAn has a n-dimensional cylindric basis itself.  E.g., a representable projective-plane Lyndon algebra with at least 6 atoms has no 5-dimensional cylindric basis.  But it clearly embeds in an atomic relation algebra with such a basis.)

12. [R Maddux] Is it decidable whether a finite relation algebra has a finite representation (i.e., with finite base)?
13. 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 n-dimensional cylindric algebra?  (It is undecidable whether a finite relation algebra is a subalgebra of the relation algebra reduct of a possibly infinite n-dimensional cylindric algebra - i.e., is in SRaCAn.)
14. [R Maddux] What is the limiting probability of a finite relation algebra being representable?

Problems of T S Ahmed:
1. Is there an uncountable atomic algebra A in Nr3CA\omega which has no complete representation?  (For countable algebras the answer is `no'.)
2. Is the intersection of Nr3CAn for finite n equal to Nr3CA\omega?
3. What can be said about the number of different CA\omegas that a given CA3 can be a neat reduct of?
Ian Hodkinson, Nov 2008, amended Mar 2012