On complete representability of Pinter's algebras and related structures

TL;DR

This paper investigates the complete representability of Pinter's algebras, establishing elementary and finitely axiomatizable classes, and explores their properties and related modal logic complexity.

Contribution

It provides new results on the axiomatizability and properties of Pinter's algebras, including their complete representability and logical complexity.

Findings

Atomic Pinter's algebras may not be completely representable.

The class of completely representable Pinter's algebras is elementary and finitely axiomatizable.

Finite dimensional multi-dimensional modal logic has NP-complete satisfiability.

Abstract

We answer an implicit question of Ian Hodkinson's. We show that atomic Pinters algebras may not be completely representable, however the class of completely representable Pinters algebras is elementary and finitely axiomatizable. We obtain analagous results for infinite dimensions (replacing finite axiomatizability by finite schema axiomatizability). We show that the class of subdirect products of set algebras is a canonical variety that is locally finite only for finite dimensions, and has the superamalgamation property; the latter for all dimensions. However, the algebras we deal with are expansions of Pinter algebras with substitutions corresponding to tranpositions. It is true that this makes the a lot of the problems addressed harder, but this is an acet, not a liability. Futhermore, the results for Pinter's algebras readily follow by just discarding the substitution operations corresponding to transpostions. Finally, we show that the multi-dimensional modal logic corresponding to finite dimensional algebras have an $NP$-complete satisfiability problem.