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* Every Boolean algebra is a distributive lattice.

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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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