Boolean algebras

Abbreviation: BA

Definition

A Boolean algebra is a structure A = (A,∨,0,∧,1,−) of type (2,0,2,0,1) such that
0,1 are identities for ∨,∧:   x∨0 = x  and  x∧1 = x
− gives a complement:   x∧−x = 0  and  x∨−x = 1
∨,∧ are associative:   x∨(y∨z) = (x∨y)∨z  and  x∧(y∧z) = (x∧y)∧z
∨,∧ are commutative:   x∨y = y∨x  and  x∧y = y∧x
∨,∧ are mutually distributive:   x∧(y∨z) = (x∧y)∨(x∧z)  and  x∨(y∧z) = (x∨y)∧(x∨z).

Definition

A Boolean algebra is a structure A = (A,∨,0,∧,1,−) of type (2,0,2,0,1) such that
(A,∨,0,∧,1) is a bounded distributive lattice,
− gives a complement:   x∧−x = 0  and  x∨−x = 1.

Morphisms

Let A and B be Boolean algebras. A morphism from A to B is a function h : A→B that is a homomorphism: h(x∨y) = h(x)∨h(y)  and  h(−x) = −h(x).

It follows that h(x∧y) = h(x)∧h(y)  and  h(0) = 0  and  h(1) = 1.

Definition

A Boolean ring is a structure A = (A, + ,0,·,1) of type (2,0,2,0) such that
(A, + ,0,·,1) is a commutative ring with unit,
· is idempotent:   x·x = x.

Remark: The term-equivalence with Boolean algebras is given by x∧y = x·y, −x = x + 1, x∨y = −(−x∧−y) and x + y = (x∨y)∧−(x∧y).

Definition

A Boolean algebra is a Heyting algebra A = (A,∨,0,∧,1,→) such that
x→0 is an involution:   ( x→0) →0 = x.

Some results

Examples

(P(S),∪,Ø,∩,S,−), the collection of subsets of a sets S, with union, intersection, and setcomplementation.

Properties

Finite members

Subclasses

Superclasses