How many varieties of cylindric algebras

TL;DR

This paper proves that there are 2^alpha varieties of geometric cylindric algebras, revealing the complexity of definable relations in infinite-variable logic and solving longstanding problems in algebraic logic.

Contribution

It introduces a new construction method to classify varieties of cylindric algebras and characterizes those generated by locally finite dimensional algebras.

Findings

There are 2^alpha varieties of geometric cylindric algebras.

Fewer varieties are generated by locally finite dimensional cylindric algebras.

Provides a recursive enumeration of all equations true of geometric cylindric algebras.

Abstract

Cylindric algebras, or concept algebras in another name, form an interface between algebra, geometry and logic; they were invented by Alfred Tarski around 1947. We prove that there are 2 to the alpha many varieties of geometric (i.e., representable) alpha-dimensional cylindric algebras, this means that 2 to the alpha properties of definable relations of (possibly infinitary) models of first-order logic theories can be expressed by formula schemes using alpha variables, where alpha is infinite. This solves Problem 4.2 in the 1985 Henkin-Monk-Tarski monograph. For solving this problem, we had to devise a new kind of construction, which we then use to solve Problem 2.13 of the 1971 Henkin-Monk-Tarski monograph which concerns the structural description of geometric cylindric algebras. There are fewer varieties generated by locally finite dimensional cylindric algebras, and we get a characterization of these among all the 2 to the alpha varieties. As a by-product, we get a simple, natural recursive enumeration of all the equations true of geometric cylindric algebras, and this can serve as a solution to Problem 4.1 of the 1985 Henkin-Monk-Tarski monograph. All this has logical content and implications concerning ordinary first order logic with a countable number of variables.