抽象代數中，同態是兩個代數結構（例如群、環、或者向量空間）之間的保持結構不變的映射。英文的同態(homomorphism)來自希臘語:μ (homos)表示"相同"而μορφ (morphe)表示"形態"。注意相似的詞根μοιο (homoios)表示"相似"出現在另一個數學概念同胚的英文(homeomorphism)中。
In abstract algebra, homomorphism is a structure-preserving mapping between two algebraic structures (such as groups, rings, or vector Spaces). English homomorphism (homomorphism) comes from the Greek: μ (homos) said the "same" and μορφ (morphe) said "form". Note similar root μοιο (homoios) said "similar" in another mathematical concept homeomorphism (homeomorphism) in English.
非正式表述
Informal expression
因為抽象代數研究帶有能產生有意義的集合上的結構或者屬性的運算的集合，最有意義的函數就是能夠保持這些運算不變的那些。它們被稱為同態。
Because abstract algebra deals with sets of operations with structures or properties that produce meaningful sets, the most meaningful functions are those that keep the operations constant. They're called homomorphisms.
例如，考慮帶加法運算的自然數。保持加法不變的函數有如下性質：f(a + b) = f(a) + f(b).例如f(x) = 3x就是這樣的一個同態，因為f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b)。注意這個同態從自然數映射回自然數。
For example, consider a natural number with an addition operation. Functions that keep addition constant have the following properties: f(a + b) = f(a) + f(b). For example, f(x) = 3x is such a homomorphism, because f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b). Notice that this homomorphism maps from a natural number back to a natural number.
同態不必從集合映射到帶相同運算的集合。例如，存在保持運算的從帶加法的實數集到帶乘法的正實數集。保持運算的函數滿足：f(a + b) = f(a) * f(b)，因為加法是第一個集合的運算而乘法是第二個集合的運算。指數定律表明f(x) = ex滿足如下條件 : 2 + 3 = 5變為e2 * e3 = e5.
Homomorphisms do not have to map from a set to a set with the same operation. For example, there is a set of real Numbers with addition to positive real Numbers with multiplication that holds operations.
Keep the function of the operation: f(a + b) = f(a) * f(b), since addition is the operation of the first set and multiplication is the operation of the second set. The exponential law states that f(x) = ex satisfies the following condition: 2 + 3 = 5 becomes e2 * e3 = e5.
同態的一個特別重要的屬性是如果么元存在，它將被保持，也即，被映射為另一個集合中的么元。注意第一個例子中f(0) = 0,而零是加法么元。第二個例子中，f(0) = 1，因為0是加法么元，而1是乘法么元。
A particularly important property of homomorphism is that if the identity exists, it will be kept, that is, mapped to the identity in another set. Notice in the first example f(0) = 0, and zero is the plus identity. In the second example, f of 0 is equal to 1, because 0 is the addition unit and 1 is the multiplication unit.
若考慮集合上的多個運算，則保持所有運算的函數可以視為同態。雖然集合相同，相同的函數可以是群論（只考慮帶一個運算的集合）中的同態，而非環論（帶兩個相關運算的集合）中的同態，因為它可能不保持環論中需要的另外那個運算。
If multiple operations on a set are considered, functions that preserve all operations can be considered homomorphic.
Although the sets are the same, the same function can be a homomorphism in a group theory (which considers only sets with one operation), rather than a homomorphism in a ring theory (which considers sets with two related operations), since it may not retain the other operation required in a ring theory.
形式化定義
Formalized definition
同態是從一個代數結構到同類代數結構的映射，它保持所有相關的結構不變；也即，所有諸如么元、逆元、和二元運算之類的屬性不變。
Homomorphismis a mapping from an algebraic structure to an algebraic structure of the same kind, which keeps all relevant structures unchanged; That is, all properties such as identity, inverse, and binary operations remain the same.
注意：有些作者在更廣的意義下使用同態一詞，而不僅是在代數中。有些人將它作為任何保持結構的映射的名稱（例如拓撲學上的連續函數），或者抽象的一般稱為範疇論中的態射的映射。本條目只考慮代數學上的同態。更廣義的用法請參看態射條目。
Note: some authors use the term homomorphism in a broader sense than just algebra. Some people use it as the name of any mapping that preserves structure (such as a topological continuous function), or an abstract mapping commonly known as morphism in category theory.
This entry considers only algebraic homomorphisms. For a more general usage, see morphism.
例如，考慮兩個有單一二元運算的集合X和Y(稱為群胚的代數結構），同態就是映射Φ:X→Y使得
For example, consider two single binary operation set X and Y (called the algebraic structure of embryo), is homomorphic mapping Φ: Φ:X→Y such that
Φ(uv)=Φ(u)○Φ(v)
其中是X上的運算而○是Y上的運算。
Where is an operation on X and ○ is an operation on Y.
每類代數結構有它的同態。同態的概念在研究所有代數結構共有的思想的泛代數中可以給一個形式化的定義。這個情況下，同態Φ:A→B是兩個同類代數結構之間的映射，使得
Each class of algebraic structure has its homomorphism. The concept of homomorphism can be formalized in general algebra, which studies the idea common to all algebraic structures. This case, the homomorphism Φ:A→B is A mapping between two similar algebraic structure, such that
對於所有n元運算F和所有A中的xi成立。
So this is true for all n factorial operations F and all xi in A.
同態的類型
The type of homomorphism
同構（isomorphism）：就是雙射的同態。兩個對象稱為同構的，如果存在相互間的同構映射。同構的對象就其上的結構而言是無法區分的。Isomorphism: a homomorphism of two morphisms. Two objects are said to be isomorphic if there is an isomorphic mapping between them. Isomorphic objects are indistinguishable in terms of the structure on top of them.滿同態（epimorphism）：就是滿射的同態。Epimorphism: that's surjective homomorphism.單同態（monomorphism）：（有時也稱擴張）是單射的同態。Monomorphism (sometimes called an expansion) is an injective homomorphism.雙同態（bimorphism）：若f既是滿同態也是單同態，則稱f為雙同態。Bimorphism: f is called a bimorphism if f is both full and single.自同態（endomorphism）：任何同態f : X → X稱為X上的一個自同態。Endomorphism: any homomorphism f: X → X is called an endomorphism on X.自同構（automorphism）：若一個自同態也是同構的，那麼稱之為自同構。Automorphism: if an automorphism is also an automorphism, it is called an automorphism.上面的術語也適用於範疇論。但是範疇論中的定義更微妙一些：細節參看態射條目。
The terms above also apply to categorism. But the categorical definition is more subtle: see morphism for details.
注意在保結構映射的意義下，定義同構為雙同態是不夠的。必須要求逆也是同類的態射。在代數意義上（至少在泛代數的意義下）這個額外的條件是自動滿足的。
Note that in the sense of structure-preserving mapping, it is not sufficient to define an isomorphism as a bihomomorphism. You have to find that the inverse is also a morphism of the same kind. In an algebraic sense (at least in a generic algebraic sense) this additional condition is automatically satisfied.
各類同態之間的關係。H = 同態的集合， M = 單同態的集合， P = 滿同態的集合， S = 同構的集合， N = 自同態的集合， A = 自同構的集合.注意: M ∩ P = S, S ∩ N = A, (M ∩ N) \ A 並且 (P ∩ N) \ A 只包含無限代數結構到自身的同態.
The relationship between homomorphisms.
H = A collection of homomorphisms， M = A collection of singlet homomorphisms，
P = A set of full homomorphisms， S = Isomorphic set，
N = A collection of endomorphisms， A = The set of automorphisms.
Note: M ∩ P = S, S ∩ N = A,(M ∩ N) \ A and (P ∩ N) \ A contains only the homomorphism of the infinite algebraic structure into itself.
同態的核
Kernel of a homomorphism
任意同態 f : X → Y 都定義了一個 X 上的等價關係 ~ 。 X 中元素 a ~ b 若且唯若 f(a) = f(b)。等價關係被稱為 f 的核。這個關係也是 X 上的一個同餘關係，因此在其商集X/~ 上也可以自然地定義一個結構：[x] * [y] = [x * y]。這時，X 通過同態 f 在 Y 中的像必然同構於 X/~。這就是所謂的同構基本定理之一。注意到在有些情況下（比如說在群結構或環結構時），僅僅一個等價類K 就可以決定商集的結構，因此這時我們可以將它記作 X/K（一般讀作 X模K ）。在這種情況下，一般將 K，而不是 ~，稱作 f 的核（參見正規子群和理想）。
Any homomorphism f: X → Y defines an equivalence relation on X ~. A to b in X if and only if f(a) = f(b). The equivalence relation is called the kernelof f.
This relation is also a congruence relation on X, so it is natural to define a structure on its quotient set X/~ : [X] * [y] = [X * y].I n this case, the image of X through the homomorphism f in Y must be isomorphic to X/~. This is one of the so-called isomorphic fundamental theorems.
Note that in some cases (such as in group or ring structures), only one equivalence class k can determine the structure of the quotient set, so we can call it X/k (usually pronounced X modulus k). In this case, k, rather than ~, is generally referred to as the kernel of f (see normal subgroups and ideals).
關係結構的同態
Homomorphism of a relational structure
模型論中，代數的結構推廣到同時涉及運算和關係的結構上。令L為由函數和關係符號組成的標識，而A,B為兩個L-結構。則從A到B的同態是映射h：從A的域到B的域，使得
In type theory, the structure of algebra is extended to structures involving both operations and relations. Let L be an identification composed of functions and relationship symbols, while A and B are two L-structures. Then the homomorphism from A to B is the mapping h: from the domain of A to the domain of B, so that
h(FA(a1,…,an)) = FB(h(a1),…,h(an))對於每個L中的n元函數符號F成立，h(FA(a1,…,an)) = FB(h(a1),…,h(an)) is true for the symbol F of n-tuple function in each L,RA(a1,…,an)推出RB(h(a1),…,h(an))對於每個L中的n元關係符號R成立。RA(a1,…,an) deducesB(h(a1),…,h(an) is true for the n-ary relation sign R in each L.在只有一個二元關係的特殊情況，這就是圖同態的概念。
In special cases where there is only one binary relationship, this is the concept of graph homomorphism.
同態和形式語言理論中的無么元同態
Homomorphism and nonmonotonic homomorphism in formal language theory
同態也被用於形式語言的研究中。給定字母表Σ和Σ，函數h : Σ* → Σ*使得h(uv)=h(u)h(v)對於所有Σ*中的u和v成立，則稱為Σ*上的同態.令e表示空詞。若h為Σ*上同態，h(x)≠e對於Σ*上所有x≠e成立，則h成為無么元同態（e-free homomorphism）。
Homomorphisms are also used in the study of formal languages. A given alphabet Σ and Σ, function h : Σ* → Σ* makes h(uv)=h(u)h(v) for all u and v in Σ*, it is called a Σ* homomorphism. e said empty words.
If h is homomorphism on Σ*, h(x)≠e holds for all x≠e on Σ*, then h will be No singlet homomorphism (e-free homomorphism).