Homomorphism

In mathematics, a homomorphism (or sometimes just a morphism) from one mathematical object to another with the same algebraic structure, is a function that preserves the operations defined on those objects.

Definition

Sean. A=(A, 1,...... , k){displaystyle {mathcal {A}}=(A,circ _{1},ldotscirc _{k})} and B=(B,↓ ↓ 1,...... ,↓ ↓ k){displaystyle {mathcal {B}}=(B,*_{1},ldots* two algebraic systems of the same type, where A,B{displaystyle A,B} are set and 1,...... , k,↓ ↓ 1,...... ,↓ ↓ k{displaystyle circ _{1},ldotscirc _{k},*_{1},ldots* are the algebraic operations defined in these sets.

A function φ φ :A→ → B{displaystyle phi:Ato B} is a homomorphism if it verifies:
φ φ ( i(a1,...... ,an))=↓ ↓ i(φ φ (a1),...... ,φ φ (an)){displaystyle phi (circ _{i}(a_{1},ldotsa_{n}))=*_{i}(phi (a_{1}),ldotsphi (a_{n})})} for each i = 1,...k and a1,...... ,an한 한 A{displaystyle a_{1},ldotsa_{n}in A}.

Examples

• The groups are sets that have defined a neutral operation and in which each element has an inverse.

So, yes. (G,↓ ↓ ),(H,⋅ ⋅ ){displaystyle (G,*), (H,cdot)} are groups, according to definition a function f:G→ → H{displaystyle f:Grightarrow H} is a homomorphism of groups if:

1. f(g1↓ ↓ g2)=f(g1)⋅ ⋅ f(g2){displaystyle f(g_{1}*g_{2})=f(g_{1})cdot f(g_{2})}} for all pairs of elements g1,g2한 한 G{displaystyle g_{1},g_{2}in G};
2. f(eG)=eH{displaystyle f(e_{G})=e_{H}}, being eG,eH{displaystyle e_{G},e_{H}}} neutral G{displaystyle G} and H{displaystyle H};
3. f(g− − 1)=f(g)− − 1{displaystyle f(g^{-1})=f(g)^{-1}}} for everything g한 한 G{displaystyle gin G}.

It can be proved that if a function satisfies the first condition then it satisfies the other two, hence the classical definition of group homomorphism does not require the other conditions.

• A K{displaystyle mathbb {K} }- vector space (where K{displaystyle mathbb {K} } is a body) is a set that has defined a sum between elements of the group and a product of climbers by elements of the set; the sum has a neutral and each element has opposite. Therefore, using the definition, for a function f:V→ → W{displaystyle f:Vto W} between two K{displaystyle mathbb {K} } vector spaces is a homomorphism must verify:

1. f(v1+v2)=f(v1)+f(v2){displaystyle f(v_{1}+v_{2})=f(v_{1})+f(v_{2})}}For everything v1,v2한 한 V{displaystyle v_{1},v_{2}in V};
2. f(λ λ ⋅ ⋅ v)=λ λ ⋅ ⋅ f(v){displaystyle f(lambda cdot v)=lambda cdot f(v)}For everything v한 한 V{displaystyle vin V} and everything λ λ 한 한 K{displaystyle lambda in mathbb {K} };
3. f(0V)=0W{displaystyle f(0_{V})=0_{W}};
4. f(− − v)=− − f(v){displaystyle f(-v)=-f(v)} for everything v한 한 V{displaystyle vin V}.

The linear transformations are exactly the functions that satisfy this (conditions 3 and 4 follow from 1 and 2). Therefore, the homomorphisms of vector spaces are the linear transformations.

• Yeah. (R,+,⋅ ⋅ ){displaystyle (R,+,cdot)} and (S,+,⋅ ⋅ ){displaystyle (S,+,cdot)} are two rings then a function f:R→ → S{displaystyle f:Rto S} is a ring homomorphism if the following two conditions are met:

1. f(a+b)=f(a)+f(b){displaystyle f(a+b)=f(a)+f(b)}Whatever. a,b한 한 R{displaystyle a,bin R};
2. f(a⋅ ⋅ b)=f(a)⋅ ⋅ f(b){displaystyle f(acdot b)=f(a)cdot f(b)}Whatever. a,b한 한 R{displaystyle a,bin R};
3. f(0R)=0S{displaystyle f(0_{R})=0_{S}};
4. f(− − a)=− − f(a){displaystyle f(-a)=-f(a)} for everything a한 한 R{displaystyle ain R}.

Conditions 3 and 4 are deduced from the first, which is why in the classical definition they are not required.

In the case of rings with unity, it is also required f(1R)=1S{displaystyle f(1_{R})=1_{S}}.

• Yeah. M{displaystyle M} and N{displaystyle N} are two R-modules (where R is a given ring) then a function f:M→ → N{displaystyle f:Mto N} It's a R-modules homomorphism if it meets the following two conditions:

1. f(m1+m2)=f(m1)+f(m2){displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}}Whatever. m1,m2한 한 M{displaystyle m_{1},m_{2}in M};
2. f(r⋅ ⋅ m)=r⋅ ⋅ f(m){displaystyle f(rcdot m)=rcdot f(m)}Whatever. m한 한 M,r한 한 R{displaystyle min M, rin R}.

Particular types of homomorphisms

• An exhaustive homomorphism is called epimorphism.
• An injective homomorphism is called monomorphism.
• A bijective homomorphism whose reverse is also a homomorphism is called isomorphism. Two objects are said to be isomorphic if there is an isomorphism of one in the other. In general, we think of two isomorphic objects as indistinguishable as to the structure in question.
• A homomorphism of a whole to itself is called endomorphism. If it is also an isomorphism is called automorphism.