Agi Kurucz

Axiomatising 2D representable diagonal-free strict cylindric algebras
(aka Non-finitely axiomatisable 2D modal product logics with infinite canonical axiomatisations)

Joint work with Christopher Hampson, Stanislav Kikot and and Sérgio Marcelino

It is well-known that the equational theory of the class RDf₂ 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 (n-dimensional cylindric relativised set algebras) in the sense that they are non-finitely axiomatisable, but have nice infinite canonical axiomatisations.

All results can also be presented in the 2D modal product logics setting. The corresponding modal logics are the first examples of non-finitely axiomatisable 2D modal product logics with infinite canonical axiomatisations, where the component logics are finitely axiomatisable.