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Boolean algebras

Abbreviation: BA nbsp nbsp nbsp nbsp nbsp Search: Boolean algebras Boolean rings

Definition

A \emph{Boolean algebra} is a structure A=A,,0,,1, of type 2,0,2,0,1 such that

0,1 are identities for ,: x0=x, x1=x

gives a complement: xx=0, xx=1

, are associative: x(yz)=(xy)z, x(yz)=(xy)z

, are commutative: xy=yx, xy=yx

, are mutually distributive: x(yz)=(xy)(xz), x(yz)=(xy)(xz)

Definition

A \emph{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: xx=0, xx=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), h(x)=h(x)

It follows that h(xy)=h(x)h(y), h(0)=0, h(1)=1.

Definition

A \emph{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: xx=x

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

Definition

A \emph{Boolean algebra} is a Heyting algebra A=A,,0,,1, such that

0 is an involution: (x0)0=x

Examples

Example 1: P(S),,,,S,, the collection of subsets of a sets S, with union, intersection, and setcomplementation.

Basic results

Properties

Finite members

Number of algebras ={1if size=2n0otherwise.

Subclasses

Superclasses

References


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