Every adjunction 〈 F, G, ε, η 〉 gives rise to an associated monad 〈 T, η, μ 〉 in the category D. The functor

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 Christmas, Victoria Square forms the centrepiece for the Frankfurt Christmas Market and Craft Fair which also extends into New Street and abuts onto a craft fair in Chamberlain Square, with the city's official Christmas tree, donated each year by Sandvik, also standing in the square.

Every element of is a member of the equivalence class.

Every two equivalence classes and are either equal or disjoint.

Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes ( or congruence classes ) for the relation.

Every homeomorphism is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent.

Every equivalence or duality of abelian categories is exact.

Every commodity is listed separately, and there is never any sign of equivalence between one unit and another.

Every ordered basis lives in one equivalence class or another.

* Every homotopy equivalence between two Hilbert manifolds is homotopic to a diffeomorphism.