Finite models are one-generated (Q1087869)

The theorem proved in this short note is: Any infinite dimensional locally finite regular cylindric set algebra with a finite base can be generated by a single element. The theorem and its proof have interesting model theoretical consequences. Firstly, every finite model is definitionally equivalent to a model with a single relation. Secondly, although on any fixed non-empty set there are infinitely many different relations, the number of definitionally non-equivalent (that is essentially different) models with the same finite universe is, in fact, finite. Finally, though a rough upper bound to the number of these models can be, and indeed is, given, the exact number is not known.