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

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 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 connected graph is an expander ; however, different connected graphs have different expansion parameters.

:"[...] Every invariant and co-variant thus becomes expressible by a graph precisely identical with a Kekuléan diagram or chemicograph.

* Every tree is a bipartite graph and a median graph.

Every tree with only countably many vertices is a planar graph.

* 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 connected graph with only countably many vertices admits a normal spanning tree.

In mathematics, Tait's conjecture states that " Every 3-connected planar cubic graph has a Hamiltonian cycle ( along the edges ) through all its vertices ".

Every directed acyclic graph has a topological ordering, an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge.

* Every planar graph whose faces all have even length is bipartite.

Every vertex of this graph has an even degree, therefore this is an Eulerian graph.

Every Eulerian orientation of a connected graph is a strong orientation, an orientation that makes the resulting directed graph strongly connected.

Every maximal outerplanar graph satisfies a stronger condition than Hamiltonicity: it is node pancyclic, meaning that for every vertex v and every k in the range from three to the number of vertices in the graph, there is a length-k cycle containing v. A cycle of this length may be found by repeatedly removing a triangle that is connected to the rest of the graph by a single edge, such that the removed vertex is not v, until the outer face of the remaining graph has length k.

Every outerplanar graph is a planar graph.

Every outerplanar graph is also a subgraph of a series-parallel graph.

Every forest, and every cactus graph is outerplanar.

Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.

They give strict interpretation to William James' statement that `` Every idea that enters the mind tends to express itself ''.

Every letter answers itself in a couple of weeks ''.

1 ) Every age rewrites the events of its history in terms of what should have been, creating legends about itself that rationalize contemporary beliefs and excuse contemporary actions.

In Norse mythology, Draupnir ( Old Norse " the dripper ") is a gold ring possessed by the god Odin with the ability to multiply itself: Every ninth night eight new rings ' drip ' from Draupnir, each one of the same size and weight as the original.

If the book itself, Animal Farm, had left any doubt of the matter, Orwell dispelled it in his essay Why I Write: ' Every line of serious work that I ’ ve written since 1936 has been written directly or indirectly against Totalitarianism ... dot, dot, dot, dot.

* Every Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )

* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

" Aside from merely disputing readings of his own work, however, in " Letter on ' Humanism ,'" Heidegger asserts that " Every humanism is either grounded in a metaphysics or is itself made to be the ground of one.

Every subgroup of a topological group is itself a topological group when given the subspace topology.

Every noun must be accompanied by the article, if any, corresponding to its definiteness, and the lack of an article ( considered a zero article ) itself specifies a certain definiteness.

" In an interesting meditation on what he viewed as the harm which would result if Christian ideals were abandoned in Belgium, he said: " Every time society has distanced itself from the Gospel, which preached humility, fraternity, and peace, the people have been unhappy, because the pagan civilization of ancient Rome, which they wanted to replace it with, is based only on pride and the abuse of force " ( Commemorative speech for the war dead of the Battle of the Yser, given by Dom Marie-Albert, Abbot of Orval Abbey, Belgium, in 1936 ).

Pope John Paul II explained in his Apostolic Letter Salvifici Doloris ( 11 February 1984 ):" In the Cross of Christ not only is the Redemption accomplished through suffering, but also human suffering itself has been redeemed ... Every man has his own share in the Redemption.

* Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer.

Every Link 16 user can identify itself to other similarly equipped platforms at ranges well beyond what Mark XII IFF systems can provide.

Every time the worshippers rose from their prayers, the mountain lifted itself upwards.

* Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval 1.

" Every sensation ," says William James, " presents itself as an indivisible unit ; and it is quite impossible to read any clear meaning into the notion that they are masses of units combined.

Every rational action must set before itself not only a principle, but also an end.

The video gained widespread media attention, with Kimmel jokingly telling the New York Times, " Every once in a while, Hollywood rallies itself for a worthy cause.

Every prayer reduces itself to this: Great God, grant that twice two be not four.

Every proposition in itself is composed out of ideas in themselves ( for simplicity, we will use proposition to mean " proposition in itself " and idea to refer to an objective idea or idea in itself.