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* Every Boolean algebra is a Heyting algebra, with given by.

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## Some Related Sentences

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__( 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__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

**.**

Every and is

__Every__legislator from Brasstown Bald to Folkston

__is__going to have his every vote subjected to the closest scrutiny as

**a**test of his political allegiances

**,**not his convictions

**.**

__Every__detail in his interpretation has been beautifully thought out

**,**and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he

__is__playing

**with**Viennese musicians )

**,**and the gossamer shading throughout

**.**

__Every__taxpayer

__is__well aware of the vast size of our annual defense budget and most of our readers also realize that

**a**large portion of these expenditures go for military electronics

**.**

__Every__few days

**,**in the early morning

**,**as the work progressed

**,**twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement

**,**supported as it

__is__

**by**ancient brick vaulting

**.**

__Every__dream

**,**and this

__is__true of

**a**mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought

**,**

__is__

**a**psychic phenomenon for which no explanation

__is__available

**.**

__Every__library borrower

**,**or at least those whose taste goes beyond the five-cent fiction rentals

**,**knows what it

__is__to hear the librarian say apologetically

**,**`` I'm sorry

**,**but we don't have that book

**.**

__Every__community

**,**if it

__is__alive has

**a**spirit

**,**and that spirit

__is__the center of its unity and identity

**.**

The restricted principle "

__Every__partially ordered set has**a**maximal totally ordered subset "__is__also equivalent to AC over ZF**.**__Every__natural-born citizen of

**a**foreign state who

__is__also an American citizen and every natural-born American citizen who

__is__

**a**citizen of

**a**foreign land owes

**a**double allegiance

**,**one to the United States

**,**and one to his homeland ( in the event of an immigrant becoming

**a**citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming

**a**citizen of another nation ).

__Every__line of written text

__is__

**a**mere reflection of references from any of

**a**multitude of traditions

**,**or

**,**as Barthes puts it

**,**" the text

__is__

**a**tissue of quotations drawn from the innumerable centres of culture "; it

__is__never original

**.**

__Every__year

**,**on the last Sunday in April

**,**there

__is__an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth

**.**

__Every__lattice element of the structure

__is__in its proper place

**,**whether it

__is__

**a**single atom or

**a**molecular grouping

**.**

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