# Boolean algebras

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### 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∨(yz) = (xy)∨z  and  x∧(yz) = (xy)∧z
∨,∧ are commutative:   xy = yx  and  xy = yx
∨,∧ are mutually distributive:   x∧(yz) = (xy)∨(xz)  and  x∨(yz) = (xy)∧(xz).

### 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 : AB that is a homomorphism: h(xy) = h(x)∨h(y)  and  h(−x) = −h(x).

It follows that h(xy) = 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 xy = x·y, x = x + 1, xy = −(−x∧−y) and x + y = (xy)∧−(xy).

### Definition

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

### Examples

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

### Properties

 Classtype variety Equational theory decidable in NPTIME Quasiequational theory decidable First-order theory decidable Congruence distributive yes Congruence modular yes Congruence n-permutable yes, n = 2 Congruence regular yes Congruence uniform yes Congruence extension property yes Definable principal congruences yes Equationally definable principal congruences yes Amalgamation property yes Strong amalgamation property yes Epimorphisms are surjective yes Locally finite yes Residual size 2

### Finite members

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  0
[Size 4]?:  1
[Size 5]?:  0
[Size 6]?:  0
[Size 7]?:  0
[Size 8]?:  1
[Size 9]?:  0
[Size 10]?:  0
[Size 11]?:  0
[Size 12]?:  0
[Size 13]?:  0
[Size 14]?:  0
[Size 15]?:  0
[Size 16]?:  1
[Size 17]?:  0
[Size 18]?:  0
[Size 19]?:  0
[Size 20]?:  0
Number of algebras  = { \begin{array}{cc} 1 & \text{if size} = 2n  0 & \text{otherwise}\end{array}.

### Subclasses

[One-element algebras]?
[Complete Boolean algebras]?

### Superclasses

Bounded distributive lattices
Generalized Boolean algebras
Heyting algebras

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