Page "Heyting algebra" ¶ 5
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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 Boolean algebra is a Heyting algebra, with given by.
* Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.
* Every Heyting algebra is a distributive lattice.

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 division ring is therefore a division algebra over its center.
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 finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.
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 state on a C *- algebra is of the above type.

Every and with
Every time I closed my eyes, I saw Gray Eyes rushing at me with a knife.
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.
Every doll in the house must be going with her ''.
** 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.
Every cladogram is based on a particular dataset that was analyzed with a particular method.
Every city expansion would imply a new circle ( canals together with town walls ).
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.
Every play is subject to booth review with coaches only having one challenge.
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.
Every song they wrote was written with an eye toward giving it " deep hidden meaning " or D. H. M.
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 year, hundreds of new titles are published in Esperanto along with music.

Every and exactly
Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:
Every member of this family does exactly what he or she wants to do.
* 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 circle in Euclidean space is a great circle of exactly one sphere.
Every positive integer appears exactly once somewhere on this list.
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 triangle has exactly three medians: one running from each vertex to the opposite side.
Every object is contained by exactly one target.
Every pair of points are i < sup > th </ sup > associates for exactly one.
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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