an induced homomorphism. A homomorphism induced from a map h {\displaystyle h} is often denoted h ∗ {\displaystyle h_{*}} . Induced homomorphisms often...

of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism. A ring homomorphism is a map between rings that preserves...

¯ c ∗ ( X , A ; G ) {\displaystyle {\bar {H}}_{c}^{*}(X,A;G)} . Induced homomorphism of this cohomology is defined as the Alexander cohomology theory...

mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function...

{\displaystyle [k]} is a ring homomorphism and is called the de Rham homomorphism. It is not difficult to show that the de Rham homomorphism is a natural transformation...

_{0}(Y)} is bijective, and for every point x in X and every n ≥ 1, the induced homomorphism f ∗ : π n ( X , x ) → π n ( Y , f ( x ) ) {\displaystyle f_{*}\colon...

be different.) On the other hand, a module homomorphism M → K is a pure embedding if the induced homomorphism between the tensor products C ⊗ M → C ⊗ K...

the (usual) Gysin homomorphism induced by the zero-section embedding X ′ ↪ N {\displaystyle X'\hookrightarrow N} . The homomorphism i! encodes intersection...

quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined). If A is...

Then, for a homomorphism f : G → H, (f(u),f(v)) is an arc (directed edge) of H whenever (u,v) is an arc of G. There is an injective homomorphism from G to...

definitions. A homomorphism of topological groups means a continuous group homomorphism G → H. Topological groups, together with their homomorphisms, form a...

properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from G ∗ H to K. Unless one of the...

substructures are precisely the induced subgraphs. However, a homomorphism between graphs is the same thing as a homomorphism between the two structures coding...

{Gray}\longrightarrow }~\operatorname {coker} c} where d is a homomorphism, known as the connecting homomorphism. Furthermore, if the morphism f is a monomorphism...

{\displaystyle f\colon X\to Y} and a point x in X, consider for any n ≥ 0 the induced homomorphism f ∗ : π n ( X , x ) → π n ( Y , f ( x ) ) , {\displaystyle f_{*}\colon...

exists a group homomorphism h ∗ : π n ( X ) → H n ( X ) , {\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X),} called the Hurewicz homomorphism, from the n-th...

∗ ( X ) {\displaystyle f^{*}\colon H^{*}(Y)\to H^{*}(X)} is the induced homomorphism in cohomology, then f ∗ ( α ⌣ β ) = f ∗ ( α ) ⌣ f ∗ ( β ) , {\displaystyle...

characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism ( R , m ) ↪ ( S , n ) {\displaystyle...

of bits Ω = 2 Z , {\displaystyle \Omega =2^{\mathbb {Z} },} the induced homomorphism is the Baker's map. Consider now the space of functions in y {\displaystyle...

isometries of (T, d), with the quotient space T/π1(Γ) = Γ. Since the induced homomorphism f# is an isomorphism between Fn = π1(Rn) and π1(Γ), we also obtain...

and with composition of loops. The resulting group homomorphism, called the induced homomorphism, is written as π ( f ) {\displaystyle \pi (f)} or, more...

dominant regular map ϕ : V → W {\displaystyle \phi \colon V\to W} , the induced homomorphism of k {\displaystyle \Bbbk } -algebras ϕ ∗ : Γ ( W ) → Γ ( V ) {\displaystyle...

f. A semigroup homomorphism between monoids preserves identity if it is a monoid homomorphism. But there are semigroup homomorphisms that are not monoid...

covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a double covering...

fundamental group of X? The answer is yes, provided we also know the induced homomorphisms π 1 ( D , ∗ ) → π 1 ( A , ∗ ) {\displaystyle \pi _{1}(D,*)\to \pi...

group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See free object. Homomorphism groups...

function f: L→M between two complete lattices L and M is a complete homomorphism if f ( ⋀ A ) = ⋀ { f ( a ) ∣ a ∈ A } {\displaystyle f\left(\bigwedge...

continuous maps between spaces induce homomorphisms between relative homology groups. In fact, this map is exactly the induced map on homology groups, but...

and (T, ∨), a homomorphism of (join-) semilattices is a function f: S → T such that f(x ∨ y) = f(x) ∨ f(y). Hence f is just a homomorphism of the two semigroups...

isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over...