Some open questions in algebraic logic

1. Is RRA axiomatisable in first-order logic with n variables, for any finite n? 

   ---

2. Is SRaCA₅ closed under completions?
(For n>5, SRaCA_n is not closed under completions.)
3. Is RA₅ closed under completions?
(For n>5, RA_n is not closed under completions.) 

---

4. Let n>4.  Is there a set of canonical equations axiomatising SRaCA_n?  Or RA_n?
5. Is there a variety of BAOs that is closed under completions (hence canonical) but with no canonical axiomatisation?
   
6. Str RCA_n is the class of atom structures of full complex algebras that are representable n-dimensional 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 non-elementary.)

   ---

7. 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?
8. Is every algebra in SRaCA_n (n>4) embeddable in an atomic relation algebra with a n-dimensional cylindric basis?
   (Not every atomic algebra in SRaCA_n 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.)

---

9. [R Maddux] Is it decidable whether a finite relation algebra has a finite representation (i.e., with finite base)?
10. 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 SRaCA_n.)
11. [R Maddux] What is the limiting probability of a finite relation algebra being representable?

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 Nr₃CA_n 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?