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 completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

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 algebraic structure has its own notion of homomorphism, namely any function compatible with the operation ( s ) defining the structure.

Every homomorphism of the Petersen graph to itself that doesn't identify adjacent vertices is an automorphism.

Every subgroup of a free abelian group is itself free abelian, which is important for the description of a general abelian group as a cokernel of a homomorphism between free abelian groups.

Every such homomorphism is of the form

Every elementary embedding is a strong homomorphism, and its image is an elementary substructure.

Every interior algebra homomorphism is a topomorphism, but not every topomorphism is an interior algebra homomorphism.

Every isogeny f: A → B is automatically a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.

Every inverse semigroup S has a E-unitary cover ; that is there exists an idempotent separating surjective homomorphism from some E-unitary semigroup T onto S.

Every locally constant function from the real numbers R to R is constant by the connectedness of R. But the function f from the rationals Q to R, defined by f ( x ) = 0 for x < π, and f ( x ) = 1 for x > π, is locally constant ( here we use the fact that π is irrational and that therefore the two sets

* Every continuous function f: → R is bounded.

Every empty function is constant, vacuously, since there are no x and y in A for which f ( x ) and f ( y ) are different when A is the empty set.

Theorem Every self-adjoint f in A * can be written as f

Every order-preserving self-map f of a cpo ( P, ⊥) has a least fixpoint.

Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.

Every morphism f: G → H in Grp has a category-theoretic kernel ( given by the ordinary kernel of algebra ker f =

Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary.

Every deterministic complexity class ( DSPACE ( f ( n )), DTIME ( f ( n )) for all f ( n )) is closed under complement, because one can simply add a last step to the algorithm which reverses the answer.

* Every point x of X is isolated in its fiber f < sup >− 1 </ sup >( f ( x )).

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

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

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 separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

* Every representable functor C → Set preserves limits ( but not necessarily colimits ).

Every universal cover p: D → X is regular, with deck transformation group being isomorphic to the fundamental group.

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

# Every subset of an independent set is independent, i. e., for each E ' ⊆ E, E ∈ I → E ' ∈ I.