Description
It is well-known that the equational theory of the class RDf_2 of 2D representable diagonal-free cylindric algebras (the algebraic counterparts of two-variable substitution and equality free first-order logic) is decidable and have a finite canonical axiomatisation. On the other hand, for n>2 the equational theory of RDF_n is not only non-finitely axiomatisable, but it does not have a canonical axiomatisation (where each equation is canonical), even if it is itself canonical and r.e. (though undecidable).
We study here a `strict' version of the cylindrifications over binary relations that correspond to the `elsewhere' quantifier in first-order logic (and so the related `rectangular' and `square' 2D representable classes both have decidable equational theories). We show that both the `rectangular' and `square' versions of 2D representable algebras of this kind behave unlike RDf_n (for n>2), rather like Crs_n in the sense that they are non-finitely axiomatisable, but have nice infinite canonical axiomatisations. We also discuss connections with 2D modal product logics.