Symmetric Boolean Algebras

TL;DR

This paper introduces Boolean algebras within a linear framework, explores their symmetric powers, and provides explicit formulas and symmetric inclusion-exclusion principles, advancing the algebraic understanding of these structures.

Contribution

It defines Boolean algebras in a linear setting, derives explicit product formulas for symmetric powers, and formulates symmetric inclusion-exclusion principles, offering new algebraic tools.

Findings

Explicit formulas for products in symmetric Boolean algebras

Symmetric forms of the inclusion-exclusion principle

Enhanced algebraic understanding of linear Boolean structures

Abstract

We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.