Powerset Boolean Algebra

Introduction

A Boolean Algebra (BA for short) is a distributive lattice with operations for conjunction, disjunction and negation. A Powerset BA is a specific case of a BA where its lattice is the powerset lattice of a set of elements called 'atoms'.

Discussion

A BA is a tuple ⟨B, ∧, ∨, ', 1, 0⟩ where

⟨B, ∧, ∨⟩ : A distributive lattice

a ∨ 0 = a and a ∧ 1 = a (for all a ∈ B)

a ∨ a' = 1 and a ∧ a' = 0 (for all a ∈ B)

The Powerset BA is also a specific case of an SCSE where

B = P(atoms) (atoms is a set of elements)

∧ = ⨆ = ∪ (or ∩ if the powerset lattice is inverted)

∨ = ⨅ = ∩ (or ∪ if inverted)

' = → ⊤

1 = ⊤ = atoms (or ∅ if inverted)

0 = ⊥ = ∅ (or atoms if inverted)

Updated:

Friday, March 04, 2016

Classes

ba

The Powerset Boolean Algebra