Independence-friendly cylindric set algebras

TL;DR

This paper introduces independence-friendly cylindric set algebras, linking them to Kleene algebras, and explores their algebraic properties and axiomatizations, advancing the algebraic understanding of independence-friendly logic.

Contribution

It defines independence-friendly cylindric set algebras and proves their connection to Kleene algebras, including finite axiomatizability of their equational theory.

Findings

Underlying Kleene algebras generate all Kleene algebras.

One-dimensional algebras have underlying monadic Kleene algebras.

The class of monadic Kleene algebras does not generate all monadic Kleene algebras.

Abstract

Independence-friendly logic is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. In her Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic called IFG logic. We attempt to algebraize IFG logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic. We define independence-friendly cylindric set algebras and prove two main results. First, every independence-friendly cylindric set algebra over a structure has an underlying Kleene algebra. Moreover, the class of such underlying Kleene algebras generates the variety of all Kleene algebras. Hence the equational theory of the class of Kleene algebras that underly an independence-friendly cylindric set algebra is finitely axiomatizable. Second, every one-dimensional independence-friendly cylindric set algebra over a structure has an underlying monadic Kleene algebra. However, the class of such underlying monadic Kleene algebras does not generate the variety of all monadic Kleene algebras. Finally, we offer a conjecture about which subvariety of monadic Kleene algebras the class of such monadic Kleene algebras does generate.