The Binary Operations of Boolean Algebras together with their Identity Elements are Monoids

A monoid (M, ∗, e) is a semigroup (M, ∗) with a two-sided identity element e. Semigroups are associative magmas and a magma (M, ∗) is just a binary operation ∗: M×M→M on M.

Given a Boolean Algebra (A, ∩, ∪, ¬, E_∪, E_∩), we want to show that (A, ∩, E_∩) and (A, ∪, E_∪) are monoids. We start by noting that ∩: A×A→A and ∪: A×A→A are binary operations. So, (A, ∩) and (A, ∪) are, by definition, magmas. Next, we note that ∩ and ∪ both satisfy the associative law. So, (A, ∩) and (A, ∪) are, by definition, semigroups.

Next, recall that the absorption and compliment axioms imply that E_∪ and E_∩ are right identities of ∪ and ∩ respectively. We can use this fact together with the commutative laws to prove that (A, ∩, E_∩) and (A, ∪, E_∪) are monoids. The proof follows.

Poof that the Right Identities of ∪ and ∩ are also Left Identities
Let a be an element of A:

1		a ∪ E∪ = a		a ∩ E∩ = a		Since E∪ and E∩ are right identities.
2		E∪ ∪ a = a		E∩ ∩ a = a		Using the Commutative Laws.
Therefore, E∪ and E∩ are Left Identities of ∪ and ∩, respectively.
Therefore, E∪ and E∩ are two-sided identities of ∪ and ∩, respectively.
Therefore, (A, ∪, E∪ ) and (A, ∩, E∩) are monoids.

We can now add left identities to our list of properties of Boolean Algebras.

Properties of Boolean Algebras

a ∪ (b ∪ c) = (a ∪ b) ∪ c		a ∩ (b ∩ c) = (a ∩ b) ∩ c		Associativity
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 = b ∪ a		a ∩ b = b ∩ a		Commutativity
E∪ ∪ a = a		E∩ ∩ a = a		Left identities
a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)		a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c)		Distributivity
Properties in blue font are axioms. Properties in black font are derived from the axioms.

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 ∩.