The Algebra of Filters of a Cubic Algebra

TL;DR

This paper explores the structure of filters in cubic algebras, introducing a congruence relation and reflection operator, thereby advancing the algebraic understanding of filter interactions within these structures.

Contribution

It introduces the concept of 'untwisted' filters leading to a new congruence relation and defines a reflection operator on filters with an associated cubic subalgebra.

Findings

'Untwisted' filters form a congruence relation.

Embeddings into interval algebras facilitate analysis.

A natural reflection operator on filters is established.

Abstract

In this paper we discuss the inclusion ordering on the filters of a filter algebra, a special type of Metropolis-Rota algeba. Using embeddings into interval algebras we show that the notion of "untwisted" gives rise to a congruence relation on the group of g-filters. We also show that there is a natural reflection operator on the class of filters with an easily definable enveloping cubic subalgebra.