[permalink] [id link]

Every ( normal ) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is isomorphic to the complex algebra corresponding to the field.

from
Wikipedia

## Some Related Sentences

Every and normal

__Every__context-sensitive grammar which does not generate

**the**empty string

**can**

**be**transformed into an equivalent one

**in**Kuroda

__normal__form

**.**

__Every__grammar

**in**Chomsky

__normal__form

**is**context-free, and conversely, every context-free grammar

**can**

**be**transformed into an equivalent one which

**is**

**in**Chomsky

__normal__form

**.**

__Every__time

**the**Flaminica saw

**a**lightning bolt

**(**Jupiter's distinctive instrument ), she was prohibited from carrying

**on**

**with**her

__normal__routine until she placated

**the**gods

**.**

__Every__first-order formula

**is**logically equivalent

**(**

**in**classical logic

**)**

**to**some formula

**in**prenex

__normal__form

**.**

__Every__first-order formula

**can**

**be**converted into Skolem

__normal__form while not changing its satisfiability via

**a**process called Skolemization

**(**sometimes spelled " Skolemnization ").

__Every__year

**the**club hosts

**a**two week long

**(**three weekends

**)**camp at Bohemian Grove, which

**is**notable for its illustrious guest list and its eclectic Cremation

**of**Care ceremony which mockingly burns " Care "

**(**

**the**

__normal__woes

**of**life

**)**

**with**grand pageantry, pyrotechnics and brilliant costumes, all done at

**the**edge

**of**

**a**lake and at

**the**base

**of**

**a**forty-foot ' stone ' owl statue

**.**

__Every__grammar

**in**Kuroda

__normal__form

**is**monotonic, and therefore, generates

**a**context-sensitive language

**.**

__Every__homeworld

**(**except Gnasty's World

**)**contains

**a**flying challenge level where Spyro's

__normal__gliding ability

**is**replaced

**with**

**the**ability

**to**fly freely

**.**

__Every__beach soccer match

**is**won by one team,

**with**

**the**game going into three minutes

**of**extra time, followed by

**a**penalty shootout if

**the**score

**is**still

**on**level terms after

__normal__time

**.**

Every and Boolean

__Every__

__Boolean__

**algebra**

**(**A, ∧, ∨) gives rise

**to**

**a**ring

**(**A, +, ·) by defining

**a**+ b :=

**(**

**a**∧ ¬ b

**)**∨

**(**b ∧ ¬

**a**

**)**=

**(**

**a**∨ b

**)**∧ ¬(

**a**∧ b

**)**

**(**this operation

**is**called symmetric difference

**in**

**the**case

**of**

**sets**and XOR

**in**

**the**case

**of**logic

**)**and

**a**· b :=

**a**∧ b

**.**The zero element

**of**this ring coincides

**with**

**the**0

**of**

**the**

__Boolean__

**algebra**;

**the**multiplicative identity element

**of**

**the**ring

**is**

**the**1

**of**

**the**

__Boolean__

**algebra**

**.**

__Every__prime ideal P

**in**

**a**

__Boolean__ring R

**is**maximal:

**the**quotient ring R / P

**is**an integral domain and also

**a**

__Boolean__ring, so

**it**

**is**

**isomorphic**

**to**

**the**

**field**F < sub > 2 </ sub >, which shows

**the**maximality

**of**P

**.**Since maximal ideals are always prime, prime ideals and maximal ideals coincide

**in**

__Boolean__rings

**.**

__Every__

__Boolean__

**algebra**

**can**

**be**obtained

**in**this way from

**a**suitable topological space: see Stone's representation theorem for

__Boolean__algebras

**.**

__Every__

__Boolean__

**algebra**

**is**

**a**Heyting

**algebra**when

**a**→ b

**is**defined

**as**usual

**as**¬

**a**∨ b,

**as**

**is**every complete distributive lattice when

**a**→ b

**is**taken

**to**

**be**

**the**supremum

**of**

**the**set

**of**all c for which

**a**∧ c ≤ b

**.**The open

**sets**

**of**

**a**topological space form

**a**complete distributive lattice and hence

**a**Heyting

**algebra**

**.**

__Every__complemented distributive lattice has

**a**unique orthocomplementation and

**is**

**in**fact

**a**

__Boolean__

**algebra**

**.**

__Every__finite

__Boolean__

**algebra**

**can**

**be**

**represented**

**as**

**a**whole power set-the power set

**of**its set

**of**atoms ; each element

**of**

**the**

__Boolean__

**algebra**corresponds

**to**

**the**set

**of**atoms below

**it**

**(**

**the**join

**of**which

**is**

**the**element ).

__Every__

__Boolean__

**algebra**A has an essentially unique completion, which

**is**

**a**complete

__Boolean__

**algebra**containing A such

**that**every element

**is**

**the**supremum

**of**some subset

**of**A

**.**

*

__Every__subset**of****a**complete__Boolean__**algebra**has**a**supremum, by definition ;**it**follows**that**every subset also has an infimum**(**greatest lower bound ).

Every and algebra

__Every__associative

__algebra__

**is**obviously alternative, but so too are some strictly nonassociative algebras such

**as**

**the**octonions

**.**

*

__Every__real Banach__algebra__which**is****a**division__algebra__**is****isomorphic****to****the**reals,**the**complexes, or**the**quaternions**.**
*

__Every__unital real Banach__algebra__**with**no zero divisors, and**in**which every principal ideal**is**closed,**is****isomorphic****to****the**reals,**the**complexes, or**the**quaternions**.**
*

__Every__commutative real unital Noetherian Banach__algebra__**with**no zero divisors**is****isomorphic****to****the**real or**complex**numbers**.**
*

__Every__commutative real unital Noetherian Banach__algebra__**(**possibly having zero divisors**)****is**finite-dimensional**.**__Every__continuous map f: X → Y induces an

__algebra__homomorphism C

**(**f

**)**: C

**(**Y

**)**→ C

**(**X

**)**by

**the**rule C

**(**f )( φ

**)**= φ o f for every φ

**in**C

**(**Y ).

__Every__vector v

**in**determines

**a**linear map from R

**to**taking 1

**to**v, which

**can**

**be**thought

**of**

**as**

**a**Lie

__algebra__homomorphism

**.**

__Every__associative

__algebra__

**is**obviously power-associative, but so are all other alternative algebras

**(**like

**the**octonions, which are non-associative

**)**and even some non-alternative algebras like

**the**sedenions

**.**

__Every__random vector gives rise

**to**

**a**probability measure

**on**R < sup > n </ sup >

**with**

**the**Borel

__algebra__

**as**

**the**underlying sigma-algebra

**.**

__Every__finite-dimensional Hausdorff topological vector space

**is**reflexive, because J

**is**bijective by linear

__algebra__, and because there

**is**

**a**unique Hausdorff vector space topology

**on**

**a**finite dimensional vector space

**.**

__Every__Heyting

__algebra__

**with**exactly one coatom

**is**subdirectly irreducible, whence every Heyting

__algebra__

**can**

**be**made an SI by adjoining

**a**new top

**.**

0.445 seconds.