On atomicity of free algebras in Boolean algebras with operators, and a new result on Pinter's free algebras

TL;DR

This paper explores the atomic structure of free algebras within Boolean algebras with operators, presenting new findings on Pinter's free algebras and revealing non-free generating sets for certain cases.

Contribution

It introduces general theorems on free algebras of Boolean algebras with operators and provides a novel result on Pinter's substitution algebras, showing non-free generating sets.

Findings

Existence of non-free generating sets for certain free algebras

New theorems on the atomicity of free algebras

Specific results for Pinter's substitution algebras

Abstract

We give some general theorems on free algebras of varieties of Boolean algebras with operators; a hitherto new result is obtained for Pinter's substitution algebras. For n\geq 3, and m>1, there is a generating set of the free algebra freely generated by m elements, which is not a free set of generators.