Building relativized representations using games

TL;DR

This paper proves key representation theorems in algebraic logic using game-theoretic methods, demonstrating complete representability of atomic algebras and properties of relativized set algebra varieties.

Contribution

It introduces game-based proofs for the Andreka-Resek-Thompson and Ferenczi representation theorems, advancing the understanding of algebraic logic structures.

Findings

Proved the Andreka-Resek-Thompson representation theorem using games.

Extended results to the polyadic analogue by Ferenczi.

Showed atomic algebras are completely representable and varieties have strong amalgamation.

Abstract

We prove the celebrated representation theorem of Andreka-Resek-Thompson, together with its polyadic analogue by Ferenczi, using games as introduced in algebraic logic by Hirsch and Hodkinson. We also show that atomic algebras are completely representable and that all such varieties of relativized set algebras have the strong amalgmation property.