Cylindric algebras and algebras of substitutions

Several new formulations of the notion of cylindric algebra are presented. The class CA of all cylindric algebras of degree a is shown to be definitionally equivalent to a class of algebras in which only substitutions (together with the Boolean +, •, and — ) are taken to be primitive operations. Then CA is shown to be definitionally equivalent to an equational class of algebras in which only substitutions and their conjugates (together with +, •, and —) are taken to be primitive operations.