Every adjunction 〈 F, G, ε, η 〉 extends an equivalence of certain subcategories.

Every monad arises from some adjunction — in fact, typically from many adjunctions — in the above fashion.

Every monad arises from some adjunction, in fact typically from many adjunctions.

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 polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).

Every statement in F #, including statements and loops, is a composable expression with a definite return type as well.

Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.

* Every subextension of F / k is separable.

* Every finite subextension of F / k is separable.

* Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra ; in fact, it is either a two-by-two matrix algebra, or a division algebra.

Every functor F: D → E induces a functor F < sup > C </ sup >: D < sup > C </ sup > → E < sup > C </ sup > ( by composition with F ).

* Every reduced commutative k-algebra A is a separable algebra ; i. e., is reduced for every field extension F / k.

President John F. Kennedy said of Hayden, " Every Federal program which has contributed to the development of the West — irrigation, power, reclamation — bears his mark, and the great Federal highway program which binds this country, together, which permits this State to be competitive east and west, north and south, this in large measure is his creation.

* Every character value is a sum of n m < sup > th </ sup > roots of unity, where n is the degree ( that is, the dimension of the associated vector space ) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer.

Every President of the United States from William Howard Taft to John F. Kennedy threw a ceremonial first pitch at least once at Griffith.

* " Every Time You Say I Love You " ( 1981 ) F & SF, May

Every sitting U. S. President since 1960 ( Eisenhower, John F. Kennedy, Lyndon B. Johnson, Richard Nixon, Gerald Ford, Jimmy Carter, Ronald Reagan, George H. W. Bush, Bill Clinton and George W. Bush ) has addressed a convention or attended an event at Cobo Center.

" Two Cars in Every Garage and Three Eyes on Every Fish " was the first episode produced for the second season, but " Bart Gets an F " aired first because of Bart's popularity and the producers wanted to premiere with an episode involving him in hopes of stealing viewers from The Cosby Show.

* Young, F. Every Man his own Mechanic ( 1881 )

* The New Dictionary of Cultural Literacy, Third Edition: What Every American Needs to Know Edited by James Trefil, Joseph F. Kett, and E. D. Hirsch.

" Every time I turn around, there's a new trial of the century ," says defense attorney F. Lee Bailey.

Every year since 1952, it has sponsored the Alfred Korzybski Memorial Lecture, with presenters from a broad range of disciplines, from science to medicine to entertainment, including names like actor Steve Allen, psychologist Albert Ellis, scientist and visionary R. Buckminster Fuller, linguist Allen Walker Read, and philosopher F. S. C. Northrop.

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.

* Every group G acts on G, i. e. in two natural but essentially different ways:, or.

Every homomorphism f: G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and.

Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G

* Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.

Every inner automorphism is indeed an automorphism of the group G, i. e. it is a bijective map from G to G and it is a homomorphism ; meaning ( xy )< sup > a </ sup >

Every non-inner automorphism yields a non-trivial element of Out ( G ), but different non-inner automorphisms may yield the same element of Out ( G ).

Water for Every Farm: A practical irrigation plan for every Australian property, K. G.

Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >.

Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if

Every year at the International Astronautical Congress, three prestigious awards are given out: the Allan D. Emil Memorial Award, the Franck J. Malina Astronautics Medal and the Luigi G. Napolitano Award.

Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological group.

Every reduced word is an alternating product of elements of G and elements of H, e. g.

* Every Polish space is homeomorphic to a G < sub > δ </ sub > subspace of the Hilbert cube, and every G < sub > δ </ sub > subspace of the Hilbert cube is Polish.

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

( 3 ) Every unitary representation of G that has an ( ε, K )- invariant unit vector for any ε > 0 and any compact subset K, has a non-zero invariant vector.

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 LORAN chain in the world uses a unique Group Repetition Interval, the number of which, when multiplied by ten, gives how many microseconds pass between pulses from a given station in the chain.

Every year, the International Documentary Festival in Amsterdam ( IDFA ) gives an acclaimed filmmaker the chance to screen his or her personal Top 10 favorite films.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

Every thing depends upon the evil of the second order ; it is this which gives to such actions the character of crime, and which makes punishment necessary.

Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.

Every ticket is provided with a bar code that gives information about the time the visitor stepped into the museum.

Every coalgebra, by ( vector space ) duality, gives rise to an algebra, but not in general the other way.

Every broadcasting company has members and the number of members gives them a status that is connected to the number of hours of broadcasting.

The WBC launched a website called Priests Rape Boys in which they criticize the Roman Catholic Church because of the Catholic sex abuse scandal, saying, " Every time any person gives any amount of money to the Catholic Church, that person is paying the salary of pedophile rapists.

Every year, the magazine gives out its Utne Independent Press Awards, which honor alternative and independent magazines from around the world.

This flat may be identified with the partition of the vertices of into the connected components of the subgraph formed by: Every set of edges having the same closure as gives rise to the same partition of the vertices, and may be recovered from the partition of the vertices, as it consists of the edges whose endpoints both belong to the same set in the partition.

Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.

Every year the American Society of Architectural Illustrators gives out the Hugh Ferriss Memorial Prize for architectural rendering excellence.

Every door, whether it leads to nothing or a level boss, gives the player a question mark ('?

Every hearts gotten gives the player-5 points.

Every trick taken gives a player 10 points.

Every group gives a quandle where the operations come from conjugation:

Every year, the IDFA, International Documentary Festival in Amsterdam, gives an acclaimed filmmaker the chance to screen his or her personal Top 10 favorite films.

Every year, she gives a speech to cadets from throughout Civil Air Patrol at the National Honor Guard Academy and also to a gathered assembly of basic cadets and staff at the Civil Air Patrol's Tri-Wing Encampment, held by Maryland Wing.

Every episode gives credits to James Burrows, Glen Charles and Les Charles, the creators of Cheers, the show where the character of Dr. Frasier Crane originated.

Every evening, she gives another twelve-minute address.

Every year the Museum gives " The Wrangler ", an original bronze sculpture by artist John Free.

Every year, Gourmand gives the awards in a very special location for gastronomy.