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Page "Distributive lattice" ¶ 12
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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 Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.
* 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 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 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 single problem touched on thus far is related to good marketing planning.
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 man in every one of these houses is a Night Rider.
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 infinite game in which is a Borel subset of Baire space is determined.
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 root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.
* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =
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.

Every and distributive
* Every Boolean algebra is a distributive lattice.
* Every totally ordered set is a distributive lattice with max as join and min as meet.
Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.
Every distributive lattice is modular.

Every and lattice
Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element.
Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.
* Every substructure is the union of its finitely generated substructures ; hence Sub ( A ) is an algebraic lattice.
Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub ( A ) for some algebra A.
Every non-modular lattice contains a copy of N < sub > 5 </ sub > as a sublattice.
Every quasinormal subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups.

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