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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 ).

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## Some Related Sentences

Every and finite

** Tukey's lemma:

__Every__non-empty collection**of**__finite__character has**a**maximal**element**with respect**to**inclusion.
Hilbert's example: "

**the**assertion that either there are only finitely many prime numbers or there are infinitely many "**(**quoted in Davis 2000: 97 ); and Brouwer's: "__Every__mathematical species**is**either__finite__or infinite.
*

__Every____finite__topological space gives rise**to****a**preorder on**its**points, in**which**x ≤ y if and only if x belongs**to**every neighborhood**of**y, and every__finite__preorder**can****be**formed**as****the**specialization preorder**of****a**topological space in this way.__Every__rational number / has two closely related expressions

**as**

**a**

__finite__continued fraction, whose coefficients

**can**

**be**determined by applying

**the**Euclidean algorithm

**to**.

__Every__

__finite__group

**of**exponent n with m generators

**is**

**a**homomorphic image

**of**B < sub > 0 </ sub >( m, n

**).**

__Every__known Sierpinski number k has

**a**small covering

**set**,

**a**

__finite__

**set**

**of**primes with at least one dividing k · 2 < sup > n </ sup >+ 1 for

**each**n > 0.

__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__oriented prime closed 3-manifold**can****be**cut along tori, so that**the**interior**of****each****of****the**resulting manifolds has**a**geometric structure with__finite__volume.__Every__

__finite__or bounded interval

**of**

**the**real numbers that contains an infinite number

**of**points must have at least one point

**of**accumulation.

__Every__field

**of**either type

**can**

**be**realized

**as**

**the**field

**of**fractions

**of**

**a**Dedekind domain in

**which**every non-zero ideal

**is**

**of**

__finite__index.

__Every__process involving charged particles emits infinitely many coherent photons

**of**infinite wavelength, and

**the**amplitude for emitting any

__finite__number

**of**photons

**is**zero.

*

__Every__finite-dimensional central simple**algebra**over**a**__finite__field must**be****a**matrix ring over that field.

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__

**(**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.

__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__Heyting

__algebra__with exactly one coatom

**is**subdirectly irreducible, whence every Heyting

__algebra__

**can**

**be**made an SI by adjoining

**a**new top.

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