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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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## Some Related Sentences

Every and Heyting

__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__totally ordered set that**is****a**bounded lattice**is**also**a**__Heyting__**algebra****,**where**is**equal to when**,**and 1 otherwise**.**

Every and algebra

__Every__associative

__algebra__

**is**obviously alternative

**,**but so too are some strictly nonassociative algebras such as the octonions

**.**

__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__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__Boolean

__algebra__

**can**

**be**obtained in this way from

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

**.**

Every and with

__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__Wednesday night through Oct. 11 there will

**be**

**an**elaborate colored fountain display

**,**

__with__229 nozzles throwing jets of water up to 130 feet

**.**

** Tukey's lemma:

__Every__non-empty collection of finite character has**a**maximal element__with__respect to inclusion**.**__Every__unit of length has

**a**corresponding unit of area

**,**namely the area of

**a**square

__with__the given side length

**.**

* Revised Code of Washington 9

**.**12**.**010: "__Every__person who brings on his or her own behalf**,**or instigates**,**incites**,**or encourages another to bring**,**any false suit at law or in equity in any court of this state**,**__with__intent thereby to distress or harass**a**defendant in the suit**,**or who serves or sends any paper or document purporting to**be**or resembling**a**judicial process**,**that**is**not in fact**a**judicial process**,****is**guilty of**a**misdemeanor ; and in case the person offending**is****an**attorney**,**he or she may**,**in addition thereto**be**disbarred from practicing law within this state**.**__Every__Hilbert space X

**is**

**a**Banach space because

**,**

**by**definition

**,**

**a**Hilbert space

**is**complete

__with__respect to the norm associated

__with__its inner product

**,**where

**a**norm and

**an**inner product are said to

**be**associated if for all x ∈ X

**.**

*

__Every__quadratic Bézier curve**is**also**a**cubic Bézier curve**,**and more generally**,****every**degree n Bézier curve**is**also**a**degree m curve for any m > n**.**In detail**,****a**degree n curve__with__control points P < sub > 0 </ sub >, …, P < sub > n </ sub >**is**equivalent ( including the parametrization ) to the degree n + 1 curve__with__control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where**.**
Jimmy Hatlo's They'll Do It

__Every__Time was often displayed in**a**two-panel format__with__the first panel showing some deceptive**,**pretentious**,**unwitting or scheming human behavior and the second panel revealing the truth of the situation**.**__Every__year

**,**the European Joint Conferences on Theory and Practice of Software ( ETAPS ) sponsors the International Conference on Compiler Construction ( CC ),

__with__papers from both the academic and industrial sectors

**.**

According to

__Every__**,****one**example may**be**" the myth of St**.**George " and other stories about saints battling dragons**,**which were " modelled no doubt in many cases on older representations of the creator and preserver of the world in combat__with__chaos ".
*

__Every__three years since Earnhardt's death**,**someone associated__with__Earnhardt has won the Daytona 500 ( his son Dale Earnhardt**,**Jr**.**in 2004**,**Earnhardt's replacement Cup driver Kevin Harvick in 2007**,**and Earnhardt Ganassi's Jamie McMurray in 2010**.**
In Norse mythology

**,**Draupnir ( Old Norse " the dripper ")**is****a**gold ring possessed**by**the god Odin__with__the ability to multiply itself:__Every__ninth night eight**new**rings ' drip ' from Draupnir**,**each**one**of the same size and weight as the original**.**__Every__system must have

**a**class designated as " root ",

__with__

**one**of its creation procedures designated as " root procedure ".

Every and exactly

* In any ring R

**,****a**maximal ideal**is****an**ideal M that**is**maximal in the set of all proper ideals of R**,**i**.**e**.**M**is**contained in__exactly__2 ideals of R**,**namely M itself and the entire ring R**.**__Every__maximal ideal**is**in fact prime**.**__Every__instance of the real-world situation or run of the experiment must produce

__exactly__

**one**outcome

**.**

__Every__year

**,**the John the Baptist's feast day reveals the cosmic dimensions of this church: at the full midday of the summer solstice

**,**when the sun

**is**at its upper culmination over the earth

**,**the light coming through the southern clerestorey windows create luminous spots that

__exactly__locate in the full midst of the nave

**with**

**a**rigorous precision

**.**

__Every__such line meets the sphere of radius

**one**centered in the origin

__exactly__twice

**,**say in P = ( x

**,**y

**,**z ) and its antipodal point (− x

**,**− y

**,**− z ).

Harrison had

**a**number of " trademark " phrases**,**such as " Morning**,**Mom ", "__Every__brand**new**day should**be**opened like**a**precious gift ", " Stay well**,**stay happy**,**stay right here " and " Harry Harrison wishing you the best ... because that's__exactly__what you deserve !” Also**,**on the last day of**every**year**,**Harrison would bring his four children to work**with**him and at the end of his shift**,**he would join them in giving listeners New Year's wishes**.**__Every__maximal outerplanar graph

**with**n vertices has

__exactly__2n − 3 edges

**,**and

**every**bounded face of

**a**maximal outerplanar graph

**is**

**a**triangle

**.**

__Every__molecule of RNA polymerase holoenzyme contains

__exactly__

**one**sigma factor subunit

**,**which in the model bacterium Escherichia coli

**is**

**one**of those listed below

**.**

#

__Every__point on line m**is**located at__exactly__the same minimum distance from line l ( equidistant lines ).
Charles W

**.**Kingston taught that “__Every__individual ... no matter what authority**,**standing**,**or station he**is**in**,****is**responsible to the**one**above him in__exactly__the same way as if that individual was the Savior himself .... We must look at the**one**above us in the same light as we look at the Savior .” This doctrine**is**known as the Law of Satisfaction and it places immense importance on honoring all participants**with**lower numbers**,**who are higher in the hierarchy**.**__Every__NFL team plays

__exactly__four pre-season exhibition games

**a**year

**,**two at home and two away

**,**

**with**the exception of two teams each year who play

**a**fifth game

**,**the Pro Football Hall of Fame Game

**.**

__Every__residue class in this group contains

__exactly__

**one**square free integer

**,**and it

**is**common

**,**therefore

**,**only to consider square free positive integers

**,**when speaking about congruent numbers

**.**

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