Closure algebras of depth two with extremal relations: Their frames, logics, and structural completeness

TL;DR

This paper investigates the properties of certain finite closure algebras with two-level extremal relations, analyzing their frames, associated logics, and structural completeness to understand their algebraic and logical characteristics.

Contribution

It introduces a detailed study of varieties generated by finite closure algebras with extremal relations, focusing on their frame structures and logical properties, including structural completeness.

Findings

Characterization of frames of depth two with extremal relations

Identification of the logical systems corresponding to these frames

Results on the structural completeness of the associated logics

Abstract

We consider varieties generated by finite closure algebras whose canonical relations have two levels, and whose restriction to a level is an "extremal" relation, i.e. the identity or the universal relation. The corresponding logics have frames of depth two, in which a level consists of a set of simple clusters or of one cluster with one or more elements.