- Homomorphism
In

abstract algebra , a**homomorphism**is a structure-preserving map between twoalgebraic structure s (such as groups, rings, orvector space s). The word "homomorphism" comes from theGreek language : "ὁμός (homos)" meaning "same" and "μορφή (morphe)" meaning "shape". Note the similar root word "ὅμοιος (homoios)", meaning "similar," which is found in another mathematical concept, namelyhomeomorphisms .**Informal discussion**Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. These functions are known as "homomorphisms".

For example, consider the

natural number s with addition as the operation. A function which preserves addition should have this property: "f"("a" + "b") = "f"("a") + "f"("b"). For example, "f"("x") = 3"x" is one such homomorphism, since "f"("a" + "b") = 3("a" + "b") = 3"a" + 3"b" = "f"("a") + "f"("b"). Note that this homomorphism maps the natural numbers back into themselves.Homomorphisms do not have to map between sets which have the same operations. For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication. A function which preserves operation should have this property: "f"("a" + "b") = "f"("a") * "f"("b"), since addition is the operation in the first set and multiplication is the operation in the second. Given the laws of

exponent s, "f"("x") = e^{"x"}satisfies this condition : 2 + 3 = 5 translates into e^{"2"}* e^{"3"}= e^{"5"}.A particularly important property of homomorphisms is that if an

identity element is present, it is always preserved, that is, mapped to the identity. Note in the first example "f"(0) = 0, and 0 is the additive identity. In the second example, "f"(0) = 1, since 0 is the additive identity, and 1 is the multiplicative identity.If we are considering multiple operations on a set, then all operations must be preserved for a function to be considered as a homomorphism. Even though the set may be the same, the same function might be a homomorphism, say, in

group theory (sets with a single operation) but not inring theory (sets with two related operations), because it fails to preserve the additional operation that ring theory considers.**Formal definition**A

**homomorphism**is a map from onealgebraic structure to another of the same type that preserves all the relevant structure; i.e. properties likeidentity element s,inverse element s, andbinary operation s.:N.B. Some authors use the word "homomorphism" in a larger context than that of algebra. Some take it to mean any kind of structure preserving map (such as continuous maps in

topology ), or even a more abstract kind of map—what we term a "morphism "—used incategory theory . This article only treats the algebraic context. For more general usage see themorphism article.For example; if one considers two sets $X$ and $Y$ with a single

binary operation defined on them (an algebraic structure known as a magma), a homomorphism is a map $phi:\; X\; ightarrow\; Y$ such that:$phi(u\; cdot\; v)\; =\; phi(u)\; circ\; phi(v)$where $cdot$ is the operation on $X$ and $circ$ is the operation on $Y$.Each type of algebraic structure has its own type of homomorphism. For specific definitions see:

*group homomorphism

*ring homomorphism

*module homomorphism

*linear operator (a homomorphism onvector space s)

*algebra homomorphism The notion of a homomorphism can be given a formal definition in the context of

universal algebra , a field which studies ideas common to allalgebraic structure s. In this setting, a homomorphism $phi:\; A\; ightarrow\; B$ is a map between two algebraic structures of the same type such that:$phi(f\_A(x\_1,\; ldots,\; x\_n))\; =\; f\_B(phi(x\_1),\; ldots,\; phi(x\_n)),$for each "n"-ary operation $f$ and for all $x\_i$ in $A$.**Types of homomorphisms*** An

is aisomorphism bijective homomorphism. Two objects are said to be isomorphic if there is an isomorphism between them. Isomorphic objects are completely indistinguishable as far as the structure in question is concerned.* An

is aepimorphism surjective homomorphism.* A

(also sometimes called an extension) is anmonomorphism injective homomorphism.* An

is a homomorphism from an object to itself.endomorphism * An

is an endomorphism which is also an isomorphism.automorphism The above terms are used in an analogous fashion in

category theory , however, the definitions incategory theory are more subtle; see the article onmorphism for more details.Note that in the larger context of structure preserving maps, it is generally insufficient to define an isomorphism as a bijective morphism. One must also require that the inverse is a morphism of the same type. In the algebraic setting (at least within the context of

universal algebra ) this extra condition is automatically satisfied.::"Relationships between different kinds of homomorphisms.

H = set of Homomorphisms, M = set of Monomorphisms,

P = set of ePimorphisms, S = set of iSomorphisms,

N = set of eNdomorphisms, A = set of Automorphisms.

Notice that: M ∩ P = S, S ∩ N = A,

(M ∩ N) A and (P ∩ N) A contain only homomorphisms from infinite algebraic structures to themselves."**Kernel of a homomorphism**Any homomorphism "f" : "X" → "Y" defines an

equivalence relation ~ on "X" by "a" ~ "b"iff "f"("a") = "f"("b"). The relation ~ is called the**kernel**of "f". It is acongruence relation on "X". Thequotient set "X"/~ can then be given an object-structure in a natural way, i.e. ["x"] * ["y"] = ["x" * "y"] . In that case the image of "X" in "Y" under the homomorphism "f" is necessarilyisomorphic to "X"/~; this fact is one of theisomorphism theorem s. Note in some cases (e.g. groups or rings), a singleequivalence class "K" suffices to specify the structure of the quotient; so we can write it "X"/"K". ("X"/"K" is usually read as "X" mod "K".) Also in these cases, it is "K", rather than ~, that is called the kernel of "f" (cf.normal subgroup , ideal).**Homomorphisms of relational structures**In

model theory , the notion of an algebraic structure is generalized to structures involving both operations and relations. Let "L" be a signature consisting of function and relation symbols, and "A", "B" be two "L"-structures. Then a**homomorphism**from "A" to "B" is a mapping "h" from the domain of "A" to the domain of "B" such that

*"h"("F"^{"A"}("a"_{1},…,"a"_{"n"})) = "F"^{"B"}("h"("a"_{1}),…,"h"("a"_{"n"})) for each "n"-ary function symbol "F" in "L",

*"R"^{"A"}("a"_{1},…,"a"_{"n"}) implies "R"^{"B"}("h"("a"_{1}),…,"h"("a"_{"n"})) for each "n"-ary relation symbol "R" in "L".In the special case with just one binary relation, we obtain the notion of agraph homomorphism .**Homomorphisms and e-free homomorphisms in formal language theory**Homomorphisms are also used in the study of formal languages. [

] Given alphabets $Sigma\_1$ and $Sigma\_2$, a function "h" : $Sigma\_1^*$ → $Sigma\_2^*$ such that $h(uv)=h(u)h(v)$ for all "u" and "v" in $Sigma\_1^*$ is called a "homomorphism" on $Sigma\_1^*$. [Seymour Ginsburg , "Algebraic and automata theoretic properties of formal languages", North-Holland, 1975, ISBN 0 7204 2506 9.*In homomorphisms on formal languages, the * operation is the*] Let "e" denote the empty word. If "h" is a homomorphism on $Sigma\_1^*$ and $h(x)\; e\; e$ for all $x\; e\; e$ in $Sigma\_1^*$, then "h" is called an "e-free homomorphism".Kleene star operation. The $cdot$ and $circ$ are bothconcatenation , commonly denoted by juxtaposition.**ee also***

morphism

*graph homomorphism

*continuous function

*homeomorphism

*diffeomorphism

*Homomorphic secret sharing - A simplistic decentralized voting protocol.**References**A monograph available free online:

* Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. " [*http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.*] " Springer-Verlag. ISBN 3-540-90578-2.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Homomorphism**— Ho mo*mor phism, n. [See {Homomorphous}.] 1. (Biol.) Same as {Homomorphy}. [1913 Webster] 2. (Bot.) The possession, in one species of plants, of only one kind of flowers; opposed to heteromorphism, dimorphism, and trimorphism. [1913 Webster] 3.… … The Collaborative International Dictionary of English**homomorphism**— [hō΄mō môr′fiz΄əm, häm΄ōmôr′fiz΄əm] n. [ HOMO + MORPH + ISM] 1. similarity in form 2. Biol. resemblance or similarity, without actual relationship, in structure or origin: said of organs or organisms 3. Bot. uniformity in shape or size, as of… … English World dictionary**homomorphism**— homomorphous, adj. /hoh meuh mawr fiz euhm, hom euh /, n. 1. Biol. correspondence in form or external appearance but not in type of structure or origin. 2. Bot. possession of perfect flowers of only one kind. 3. Zool. resemblance between the… … Universalium**homomorphism**— homomorfizmas statusas T sritis fizika atitikmenys: angl. homomorphism vok. Homomorphismus, m rus. гомоморфизм, m pranc. homomorphisme, m … Fizikos terminų žodynas**homomorphism**— noun Etymology: International Scientific Vocabulary Date: 1935 a mapping of a mathematical set (as a group, ring, or vector space) into or onto another set or itself in such a way that the result obtained by applying the operations to elements of … New Collegiate Dictionary**homomorphism**— noun a) A structure preserving map between two algebraic structures, such as groups, rings, or vector spaces. b) A similar appearance of two unrelated organisms or structures See Also: morphism … Wiktionary**homomorphism**— homo·mor·phism … English syllables**homomorphism**— ho•mo•mor•phism [[t]ˌhoʊ məˈmɔr fɪz əm, ˌhɒm ə [/t]] also ho′mo•mor phy n. 1) dvl correspondence in form or external appearance 2) bot possession of perfect flowers of only one kind • Etymology: 1865–70 ho mo•mor′phous, ho mo•mor′phic, adj … From formal English to slang**homomorphism**— /hoʊmoʊˈmɔfɪzəm/ (say hohmoh mawfizuhm) noun 1. Biology correspondence in form or external appearance but not in type of structure or in origin. 2. Zoology resemblance between the young and the adult. Also, homomorphy. {homo + morph + ism}… … Australian English dictionary**homomorphism**— noun similarity of form • Syn: ↑homomorphy • Hypernyms: ↑similarity … Useful english dictionary