Formalizing first-order logic in the equational theory of 3-dimensional diagonal-free cylindric algebras (Q271687)

The following conjecture of \textit{I. Németi} [``Logic with three variables has Gödel's incompleteness property'', Preprint] is proved: if the parameters \(e\) and \(p\) of a diagonal free cylindic algebra \((\mathrm{Df}_3)\) satisfy certain equations (denoted by \(\mathrm{Ax}\)), then the relation algebra reduct of this \(\mathrm{Df}_3\) results a quasi-projective relation algebra. Furthermore, using the previous theorem, a recursive function translating the sentences of first order logic to the equational theory of \(\mathrm{Df}_3\) is defined, removing the relativization to set theory, strengthening a result of \textit{H. Andréka} and \textit{I. Németi} [``Formalizing set theory in diagonal-free cylindric algebras, searching for the weakest logic with Gödel's incompleteness property'', Preprint, \url{http://www.renyi.hu/ nemeti/NDis/diagonalfree.pdf}].