Atom structures of cylindric algebras and relation algebras
In 1970, \textit{J. D. Monk} [Math. Nachr. 46, 47-55 (1970; Zbl 0182.32301)] proved that relation algebras and cylindric algebras have completions, and asked whether completions of representable algebras are representable. This paper shows the answer is ``no''. There is a countable infinite relational structure \({\mathfrak S}\) whose (uncountable) complex algebra is a relation algebra that is not representable, and yet \({\mathfrak S}\) is the atom structure of a countable representable relation algebra. It follows that there is a countable atomic representable relation algebra whose completion is not representable. Similar results hold for cylindric algebras and solve a problem of \textit{L. Henkin, J. D. Monk}, and \textit{A. Tarski} [Cylindric algebras. Part I (1971; Zbl 0214.01302), Remarks 2.7.25 and Problem 2.16] by showing that the classes of representable relation algebras and representable cylindric algebras of finite dimension \(n\) \((n\geq 3)\) are not axiomatizable by equations which are ``positive in the wider sense''.
scientific article; zbMATH DE number 1109906
scientific article