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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.
Some Related Sentences
Every and normal
Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form.
Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form.
Every time the Flaminica saw a lightning bolt ( Jupiter's distinctive instrument ), she was prohibited from carrying on with her normal routine until she placated the gods.
Every first-order formula is logically equivalent ( in classical logic ) to some formula in prenex normal form.
Every first-order formula can be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization ( sometimes spelled " Skolemnization ").
Every year the club hosts a two week long ( three weekends ) camp at Bohemian Grove, which is notable for its illustrious guest list and its eclectic Cremation of Care ceremony which mockingly burns " Care " ( the normal woes of life ) with grand pageantry, pyrotechnics and brilliant costumes, all done at the edge of a lake and at the base of a forty-foot ' stone ' owl statue.
Every grammar in Kuroda normal form is monotonic, and therefore, generates a context-sensitive language.
Every homeworld ( except Gnasty's World ) contains a flying challenge level where Spyro's normal gliding ability is replaced with the ability to fly freely.
Every beach soccer match is won by one team, with the game going into three minutes of extra time, followed by a penalty shootout if the score is still on level terms after normal time.
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 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.