The Distinguished Elements of Boolean Algebra are Identities

The absorption and compliment axioms of Boolean Algebra imply that the distinguished elements e_∩ and e_∪ are right identities of the conjunction and disjunction operations. Here are poofs: Let (A, ∩, ∪, ¬, e_∩, e_∪) be a Boolean algebra. Let a be an element of A. Let b = ¬a and define E_∩ = e_∪ and E_∪ = e_∩.

Proof that E∪ is a right identity of the disjunction operation ∪
a ∪ (a ∩ b) = a	Absorption axiom.	1
b = ¬a	By definition.	2
a ∪ (a ∩ ¬a) = a	By 1 and 2.	3
a ∩ ¬a = e∩	Complement axiom.	4
a ∪ e∩ = a	By 3 and 4.	5
E∪ = e∩	By definition.	6
a ∪ E∪ = a	By 5 and 6.	7
Therefore, E∪ is a right identity of the disjunction operation ∪.

Proof that E∩ is a right identity of the conjunction operation ∩
a ∩ (a ∪ b) = a	Absorption axiom.	1
b = ¬a	By definition.	2
a ∩ (a ∪ ¬a) = a	By 1 and 2.	3
a ∪ ¬a = e∪	Complement axiom.	4
a ∩ e∪ = a	By 3 and 4.	5
E∩ = e∪	By definition.	6
a ∩ E∩ = a	By 5 and 6.	7
Therefore, E∩ is a right identity of the conjunction operation ∩.

We can now add the existence of right identities to our list of properties of Boolean Algebras.

A Boolean Algebra is a 6-tuple (A, ∩, ∪, ¬, E∪, E∩) that satisfies the following Properties For all a, b, and c in A:
a ∪ (a ∩ b) = a	a ∩ (a ∪ b) = a	Absorption
a ∪ ¬a = E∩	a ∩ ¬a = E∪	Complements
a ∪ E∪ = a	a ∩ E∩ = a	Right identities
a ∪ (b ∪ c) = (a ∪ b) ∪ c	a ∩ (b ∩ c) = (a ∩ b) ∩ c	Associativity
a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)	a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)	Distributivity
a ∪ b = b ∪ a	a ∩ b = b ∩ a	Commutativity
Axioms are in blue font. Derived properties, just one of them so far—the existence of right identities—are listed in black font.

Notice that I have replaced e_∩ and e_∪ with E_∪ and E_∩ respectively. The ¬ operator is not a true analog to negation. Since a ∪ ¬a = E_∩ (the identity of ∩) rather than E_∪ (the identity of ∪), ¬a is not an inverse of a with respect to ∪. Likewise, ¬a is not and inverse of a with respect to ∩.

The Standard Axioms of Boolean Algebra

A Boolean algebra is a 6-tuple consisting of a set A, equipped with two binary operations, a unary operation, and two distinguished elements. The first binary operation ∩: A × A → A is referred to as “conjunction”, “meet”, or “and”. The second binary operation ∪: A × A → A is referred to as “disjunction”, “join”, or “or”. The unary operation ¬: A → A is referred to as “compliment” or “not”. The distinguished elements are often denoted as 0 and 1. However, I will denote these elements as e_∩ and e_∪.

The 6-tuple (A, ∩, ∪, ¬, e_∩, e_∪) is called a “Boolean algebra” if it satisfies certain stated axioms. The axioms differ depending upon the needs of a particular author. The five following axiom pairs are a particularly common formulation of Boolean algebra. For all a, b, and c in A:

The Standard Axioms of Boolean Algebra
a ∪ (a ∩ b) = a	a ∩ (a ∪ b) = a	absorption
a ∪ (b ∪ c) = (a ∪ b) ∪ c	a ∩ (b ∩ c) = (a ∩ b) ∩ c	associativity
a ∪ ¬a = e∪	a ∩ ¬a = e∩	complements
a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)	a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)	distributivity
a ∪ b = b ∪ a	a ∩ b = b ∩ a	commutativity