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Injective function

In mathematics, an injective function (also known as injection, or one-to-one function ) is a function $f$ that maps distinct elements of its domain to distinct elements; that is, $x_{1} ≠ x_{2}$ implies $f(x_{1}) f(x_{2})$. (Equivalently, $f(x_{1}) = f(x_{2})$ implies $x_{1} = x_{2}$ in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with  that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details.

A function $$f$$ that is not injective is sometimes called many-to-one.

Definition

Let $$f$$ be a function whose domain is a set $$X.$$ The function $$f$$ is said to be injective provided that for all $$a$$ and $$b$$ in $$X,$$ if $$f(a) = f(b),$$ then $$a = b$$; that is, $$f(a) = f(b)$$ implies $$a=b.$$ Equivalently, if $$a \neq b,$$ then $$f(a) \neq f(b)$$ in the contrapositive statement.

Symbolically,$$\forall a,b \in X, \;\; f(a)=f(b) \Rightarrow a=b,$$ which is logically equivalent to the contrapositive, $$\forall a, b \in X, \;\; a \neq b \Rightarrow f(a) \neq f(b).$$

Examples
For visual examples, readers are directed to the gallery section.
 * For any set $$X$$ and any subset $$S \subseteq X,$$ the inclusion map $$S \to X$$ (which sends any element $$s \in S$$ to itself) is injective. In particular, the identity function $$X \to X$$ is always injective (and in fact bijective).
 * If the domain of a function is the empty set, then the function is the empty function, which is injective.
 * If the domain of a function has one element (that is, it is a singleton set), then the function is always injective.
 * The function $$f : \R \to \R$$ defined by $$f(x) = 2 x + 1$$ is injective.
 * The function $$g : \R \to \R$$ defined by $$g(x) = x^2$$ is injective, because (for example) $$g(1) = 1 = g(-1).$$ However, if $$g$$ is redefined so that its domain is the non-negative real numbers [0,+∞), then $$g$$ is injective.
 * The exponential function $$\exp : \R \to \R$$ defined by $$\exp(x) = e^x$$ is injective (but not surjective, as no real value maps to a negative number).
 * The natural logarithm function $$\ln : (0, \infty) \to \R$$ defined by $$x \mapsto \ln x$$ is injective.
 * The function $$g : \R \to \R$$ defined by $$g(x) = x^n - x$$ is not injective, since, for example, $$g(0) = g(1) = 0.$$

More generally, when $$X$$ and $$Y$$ are both the real line $$\R,$$ then an injective function $$f : \R \to \R$$ is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the.

Injections can be undone
Functions with left inverses are always injections. That is, given $$f : X \to Y,$$ if there is a function $$g : Y \to X$$ such that for every $$x \in X$$, $$g(f(x)) = x$$, then $$f$$ is injective. In this case, $$g$$ is called a retraction of $$f.$$ Conversely, $$f$$ is called a section of $$g.$$

Conversely, every injection $$f$$ with a non-empty domain has a left inverse $$g$$. It can be defined by choosing an element $$a$$ in the domain of $$f$$ and setting $$g(y)$$ to the unique element of the pre-image $$f^{-1}[y]$$ (if it is non-empty) or to $$a$$ (otherwise).

The left inverse $$g$$ is not necessarily an inverse of $$f,$$ because the composition in the other order, $$f \circ g,$$ may differ from the identity on $$Y.$$ In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.

Injections may be made invertible
In fact, to turn an injective function $$f : X \to Y$$ into a bijective (hence invertible) function, it suffices to replace its codomain $$Y$$ by its actual image $$J = f(X).$$ That is, let $$g : X \to J$$ such that $$g(x) = f(x)$$ for all $$x \in X$$; then $$g$$ is bijective. Indeed, $$f$$ can be factored as $$\operatorname{In}_{J,Y} \circ g,$$ where $$\operatorname{In}_{J,Y}$$ is the inclusion function from $$J$$ into $$Y.$$

More generally, injective partial functions are called partial bijections.

Other properties

 * If $$f$$ and $$g$$ are both injective then $$f \circ g$$ is injective.
 * If $$g \circ f$$ is injective, then $$f$$ is injective (but $$g$$ need not be).
 * $$f : X \to Y$$ is injective if and only if, given any functions $$g,$$ $$h : W \to X$$ whenever $$f \circ g = f \circ h,$$ then $$g = h.$$ In other words, injective functions are precisely the monomorphisms in the category Set of sets.
 * If $$f : X \to Y$$ is injective and $$A$$ is a subset of $$X,$$ then $$f^{-1}(f(A)) = A.$$ Thus, $$A$$ can be recovered from its image $$f(A).$$
 * If $$f : X \to Y$$ is injective and $$A$$ and $$B$$ are both subsets of $$X,$$ then $$f(A \cap B) = f(A) \cap f(B).$$
 * Every function $$h : W \to Y$$ can be decomposed as $$h = f \circ g$$ for a suitable injection $$f$$ and surjection $$g.$$ This decomposition is unique up to isomorphism, and $$f$$ may be thought of as the inclusion function of the range $$h(W)$$ of $$h$$ as a subset of the codomain $$Y$$ of $$h.$$
 * If $$f : X \to Y$$ is an injective function, then $$Y$$ has at least as many elements as $$X,$$ in the sense of cardinal numbers. In particular, if, in addition, there is an injection from $$Y$$ to $$X,$$ then $$X$$ and $$Y$$ have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
 * If both $$X$$ and $$Y$$ are finite with the same number of elements, then $$f : X \to Y$$ is injective if and only if $$f$$ is surjective (in which case $$f$$ is bijective).
 * An injective function which is a homomorphism between two algebraic structures is an embedding.
 * Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function $$f$$ is injective can be decided by only considering the graph (and not the codomain) of $$f.$$

Proving that functions are injective
A proof that a function $$f$$ is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if $$f(x) = f(y),$$ then $$x = y.$$

Here is an example: $$f(x) = 2 x + 3$$

Proof: Let $$f : X \to Y.$$ Suppose $$f(x) = f(y).$$  So $$2 x + 3 = 2 y + 3$$ implies $$2 x = 2 y,$$ which implies $$x = y.$$  Therefore, it follows from the definition that $$f$$ is injective.

There are multiple other methods of proving that a function is injective. For example, in calculus if $$f$$ is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if $$f$$ is a linear transformation it is sufficient to show that the kernel of $$f$$ contains only the zero vector. If $$f$$ is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.

A graphical approach for a real-valued function $$f$$ of a real variable $$x$$ is the horizontal line test. If every horizontal line intersects the curve of $$f(x)$$ in at most one point, then $$f$$ is injective or one-to-one.

title: Injective function


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Inverse element

In mathematics, the concept of an inverse element generalises the concepts of opposite ($−x$) and reciprocal ($1/x$) of numbers.

Given an operation denoted here $∗$, and an identity element denoted $e$, if $x ∗ y = e$, one says that $x$ is a left inverse of $y$, and that $y$ is a right inverse of $x$. (An identity element is an element such that $x * e = x$ and $e * y = y$ for all $x$ and $y$ for which the left-hand sides are defined. )

When the operation $∗$ is associative, if an element $x$ has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the inverse element or simply the inverse. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an invertible element, also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under addition).

Inverses are commonly used in groups—where every element is invertible, and rings—where invertible elements are also called units. They are also commonly used for operations that are not defined for all possible operands, such as inverse matrices and inverse functions. This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism.

The word 'inverse' is derived from inversus that means 'turned upside down', 'overturned'. This may take its origin from the case of fractions, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of $$\tfrac x y$$ is $$\tfrac y x$$).

Definitions and basic properties
The concepts of inverse element and invertible element are commonly defined for binary operations that are everywhere defined (that is, the operation is defined for any two elements of its domain). However, these concepts are also commonly used with partial operations, that is operations that are not defined everywhere. Common examples are matrix multiplication, function composition and composition of morphisms in a category. It follows that the common definitions of associativity and identity element must be extended to partial operations; this is the object of the first subsections.

In this section, $X$ is a set (possibly a proper class) on which a partial operation (possibly total) is defined, which is denoted with $$*.$$

Associativity
A partial operation is associative if
 * $$x*(y*z)=(x*y)*z$$

for every $x, y, z$ in $X$ for which one of the members of the equality is defined; the equality means that the other member of the equality must also be defined.

Examples of non-total associative operations are multiplication of matrices of arbitrary size, and function composition.

Identity elements
Let $$*$$ be a possibly partial associative operation on a set $X$.

An identity element, or simply an identity is an element $e$ such that
 * $$x*e=x \quad\text{and}\quad e*y=y$$

for every $x$ and $y$ for which the left-hand sides of the equalities are defined.

If $e$ and $f$ are two identity elements such that $$e*f$$ is defined, then $$e=f.$$ (This results immediately from the definition, by $$e=e*f=f.$$)

It follows that a total operation has at most one identity element, and if $e$ and $f$ are different identities, then $$e*f$$ is not defined.

For example, in the case of matrix multiplication, there is one $n×n$ identity matrix for every positive integer $n$, and two identity matrices of different size cannot be multiplied together.

Similarly, identity functions are identity elements for function composition, and the composition of the identity functions of two different sets are not defined.

Left and right inverses
If $$x*y=e,$$ where $e$ is an identity element, one says that $x$ is a left inverse of $y$, and $y$ is a right inverse of $x$.

Left and right inverses do not always exist, even when the operation is total and associative. For example, addition is a total associative operation on nonnegative integers, which has $0$ as additive identity, and $0$ is the only element that has an additive inverse. This lack of inverses is the main motivation for extending the natural numbers into the integers.

An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the functions from the integers to the integers. The doubling function $$x\mapsto 2x$$ has infinitely many left inverses under function composition, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps $n$ to either $$2n$$ or $$2n+1$$ is a right inverse of the function $n\mapsto \left\lfloor \frac n2\right\rfloor,$ the floor function that maps $n$ to $\frac n2$  or $\frac{n-1}2,$  depending whether $n$ is even or odd.

More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is surjective.

In category theory, right inverses are also called sections, and left inverses are called retractions.

Inverses
An element is invertible under an operation if it has a left inverse and a right inverse.

In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if $l$ and $r$ are respectively a left inverse and a right inverse of $x$, then
 * $$l=l*(x*r)=(l*x)*r=r.$$

The inverse of an invertible element is its unique left or right inverse.

If the operation is denoted as an addition, the inverse, or additive inverse, of an element $x$ is denoted $$-x.$$ Otherwise, the inverse of $x$ is generally denoted $$x^{-1},$$ or, in the case of a commutative multiplication \frac 1x. When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in $$x^{* -1}.$$ The notation $$f^{\circ -1}$$ is not commonly used for function composition, since \frac 1f can be used for the multiplicative inverse.

If $x$ and $y$ are invertible, and $$x*y$$ is defined, then $$x*y$$ is invertible, and its inverse is $$y^{-1}x^{-1}.$$

An invertible homomorphism is called an isomorphism. In category theory, an invertible morphism is also called an isomorphism.

In groups
A group is a set with an associative operation that has an identity element, and for which every element has an inverse.

Thus, the inverse is a function from the group to itself that may also be considered as an operation of arity one. It is also an involution, since the inverse of the inverse of an element is the element itself.

A group may act on a set as transformations of this set. In this case, the inverse $$g^{-1}$$ of a group element $$g$$ defines a transformation that is the inverse of the transformation defined by $$g,$$ that is, the transformation that "undoes" the transformation defined by $$g.$$

For example, the Rubik's cube group represents the finite sequences of elementary moves. The inverse of such a sequence is obtained by applying the inverse of each move in the reverse order.

In monoids
A monoid is a set with an associative operation that has an identity element.

The invertible elements in a monoid form a group under monoid operation.

A ring is a monoid for ring multiplication. In this case, the invertible elements are also called units and form the group of units of the ring.

If a monoid is not commutative, there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible).

For example, the set of the functions from a set to itself is a monoid under function composition. In this monoid, the invertible elements are the bijective functions; the elements that have left inverses are the injective functions, and those that have right inverses are the surjective functions.

Given a monoid, one may want extend it by adding inverse to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the cancellation property (an element $x$ has the cancellation property if $$xy=xz$$ implies $$y=z,$$ and $$yx=zx$$ implies $y=z$). This extension of a monoid is allowed by Grothendieck group construction. This is the method that is commonly used for constructing integers from natural numbers, rational numbers from integers and, more generally, the field of fractions of an integral domain, and localizations of commutative rings.

In rings
A ring is an algebraic structure with two operations, addition and multiplication, which are denoted as the usual operations on numbers.

Under addition, a ring is an abelian group, which means that addition is commutative and associative; it has an identity, called the additive identity, and denoted $0$; and every element $x$ has an inverse, called its additive inverse and denoted $−x$. Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses.

Under multiplication, a ring is a monoid; this means that multiplication is associative and has an identity called the multiplicative identity and denoted $1$. An invertible element  for multiplication is called a unit. The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit $x$ is denoted $$x^{-1},$$ or, when the multiplication is commutative, $\frac 1x.$

The additive identity $0$ is never a unit, except when the ring is the zero ring, which has $0$ as its unique element.

If $0$ is the only non-unit, the ring is a field if the multiplication is commutative, or a division ring otherwise.

In a noncommutative ring (that is, a ring whose multiplication is not commutative), a non-invertible element may have one or several left or right inverses. This is, for example, the case of the linear functions from a infinite-dimensional vector space to itself.

A commutative ring (that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not zero divisors (that is, their product with a nonzero element cannot be $0$). This is the process of localization, which produces, in particular, the field of rational numbers from the ring of integers, and, more generally, the field of fractions of an integral domain. Localization is also used with zero divisors, but, in this case the original ring is not a subring of the localisation; instead, it is mapped non-injectively to the localization.

Matrices
Matrix multiplication is commonly defined for matrices over a field, and straightforwardly extended to matrices over rings, rngs and semirings. However, in this section, only matrices over a commutative ring are considered, because of the use of the concept of rank and determinant.

If $A$ is a $m×n$ matrix (that is, a matrix with $m$ rows and $n$ columns), and $B$ is a $p×q$ matrix, the product $AB$ is defined if $n = p$, and only in this case. An identity matrix, that is, an identity element for matrix multiplication is a square matrix (same number for rows and columns) whose entries of the main diagonal are all equal to $1$, and all other entries are $0$.

An invertible matrix is an invertible element under matrix multiplication. A matrix over a commutative ring $R$ is invertible if and only if its determinant is a unit in $R$ (that is, is invertible in $R$. In this case, its inverse matrix can be computed with Cramer's rule.

If $R$ is a field, the determinant is invertible if and only if it is not zero. As the case of fields is more common, one see often invertible matrices defined as matrices with a nonzero determinant, but this is incorrect over rings.

In the case of integer matrices (that is, matrices with integer entries), an invertible matrix is a matrix that has an inverse that is also an integer matrix. Such a matrix is called a unimodular matrix for distinguishing it from matrices that are invertible over the real numbers. A square integer matrix is unimodular if and only if its determinant is $1$ or $−1$, since these two numbers are the only units in the ring of integers.

A matrix has a left inverse if and only if its rank equals its number of columns. This left inverse is not unique except for square matrices where the left inverse equal the inverse matrix. Similarly, a right inverse exists if and only if the rank equals the number of rows; it is not unique in the case of a rectangular matrix, and equals the inverse matrix in the case of a square matrix.

Functions, homomorphisms and morphisms
Composition is a partial operation that generalizes to homomorphisms of algebraic structures and morphisms of categories into operations that are also called composition, and share many properties with function composition.

In all the case, composition is associative.

If $$f\colon X\to Y$$ and $$g\colon Y'\to Z,$$ the composition $$g\circ f$$ is defined if and only if $$Y'=Y$$ or, in the function and homomorphism cases, $$Y\subset Y'.$$ In the function and homomorphism cases, this means that the codomain of $$f$$ equals or is included in the domain of $g$. In the morphism case, this means that the codomain of $$f$$ equals the domain of $g$.

There is an identity $$\operatorname{id}_X \colon X \to X$$ for every object $X$ (set, algebraic structure or object), which is called also an identity function in the function case.

A function is invertible if and only if it is a bijection. An invertible homomorphism or morphism is called an isomorphism. An homomorphism of algebraic structures is an isomorphism if and only if it is a bijection. The inverse of a bijection is called an inverse function. In the other cases, one talks of inverse isomorphisms.

A function has a left inverse or a right inverse if and only it is injective or surjective, respectively. An homomorphism of algebraic structures that has a left inverse or a right inverse is respectively injective or surjective, but the converse is not true in some algebraic structures. For example, the converse is true for vector spaces but not for modules over a ring: a homomorphism of modules that has a left inverse of a right inverse is called respectively a split epimorphism or a split monomorphism. This terminology is also used for morphisms in any category.

In a unital magma
Let $$S$$ be a unital magma, that is, a set with a binary operation $$*$$ and an identity element $$e\in S$$. If, for $$a,b\in S$$, we have $$a*b=e$$, then $$a$$ is called a left inverse of $$b$$ and $$b$$ is called a right inverse of $$a$$. If an element $$x$$ is both a left inverse and a right inverse of $$y$$, then $$x$$ is called a two-sided inverse, or simply an inverse, of $$y$$. An element with a two-sided inverse in $$S$$ is called invertible in $$S$$. An element with an inverse element only on one side is left invertible or right invertible.

Elements of a unital magma $$(S,*)$$ may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table the elements 2 and 3 each have two two-sided inverses.

A unital magma in which all elements are invertible need not be a loop. For example, in the magma $$(S,*)$$ given by the Cayley table every element has a unique two-sided inverse (namely itself), but $$(S,*)$$ is not a loop because the Cayley table is not a Latin square.

Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table the only element with a two-sided inverse is the identity element 1.

If the operation $$*$$ is associative then if an element has both a left inverse and a right inverse, they are equal. In other words, in a monoid (an associative unital magma) every element has at most one inverse (as defined in this section). In a monoid, the set of invertible elements is a group, called the group of units of $$S$$, and denoted by $$U(S)$$ or H1.

In a semigroup
The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity; that is, in a semigroup.

In a semigroup S an element x is called (von Neumann) regular if there exists some element z in S such that xzx = x; z is sometimes called a pseudoinverse. An element y is called (simply) an inverse of x if xyx = x and y = yxy. Every regular element has at least one inverse: if x = xzx then it is easy to verify that y = zxz is an inverse of x as defined in this section. Another easy to prove fact: if y is an inverse of x then e = xy and f = yx are idempotents, that is ee = e and ff = f. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and ex = xf = x, ye = fy = y, and e acts as a left identity on x, while f acts a right identity, and the left/right roles are reversed for y. This simple observation can be generalized using Green's relations: every idempotent e in an arbitrary semigroup is a left identity for Re and right identity for Le. An intuitive description of this fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.

In a monoid, the notion of inverse as defined in the previous section is strictly narrower than the definition given in this section. Only elements in the Green class H1 have an inverse from the unital magma perspective, whereas for any idempotent e, the elements of He have an inverse as defined in this section. Under this more general definition, inverses need not be unique (or exist) in an arbitrary semigroup or monoid. If all elements are regular, then the semigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly one inverse as defined in this section, then the semigroup is called an inverse semigroup. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an absorbing element 0 because 000 = 0, whereas a group may not.

Outside semigroup theory, a unique inverse as defined in this section is sometimes called a quasi-inverse. This is generally justified because in most applications (for example, all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity (see Generalized inverse).

U-semigroups
A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (a°)° = a for all a in S; this endows S with a type ⟨2,1⟩ algebra. A semigroup endowed with such an operation is called a U-semigroup. Although it may seem that a° will be the inverse of a, this is not necessarily the case. In order to obtain interesting notion(s), the unary operation must somehow interact with the semigroup operation. Two classes of U-semigroups have been studied:

 * I-semigroups, in which the interaction axiom is aa°a = a
 * *-semigroups, in which the interaction axiom is (ab)° = b°a°. Such an operation is called an involution, and typically denoted by a*

Clearly a group is both an I-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are completely regular semigroups; these are I-semigroups in which one additionally has aa° = a°a; in other words every element has commuting pseudoinverse a°. There are few concrete examples of such semigroups however; most are completely simple semigroups. In contrast, a subclass of *-semigroups, the *-regular semigroups (in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the Moore–Penrose inverse. In this case however the involution a* is not the pseudoinverse. Rather, the pseudoinverse of x is the unique element y such that xyx = x, yxy = y,  (xy)* = xy, (yx)* = yx. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the generalized inverse or Moore–Penrose inverse.

Examples
All examples in this section involve associative operators.

Galois connections
The lower and upper adjoints in a (monotone) Galois connection, L and G are quasi-inverses of each other; that is, LGL = L and GLG = G and one uniquely determines the other. They are not left or right inverses of each other however.

Generalized inverses of matrices
A square matrix $$M$$ with entries in a field $$K$$ is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero. If the determinant of $$M$$ is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See invertible matrix for more.

More generally, a square matrix over a commutative ring $$R$$ is invertible if and only if its determinant is invertible in $$R$$.

Non-square matrices of full rank have several one-sided inverses:
 * For $$A:m\times n \mid m>n$$ we have left inverses; for example, $$\underbrace{ \left(A^\text{T}A\right)^{-1} A^\text{T} }_{ A^{-1}_\text{left} } A = I_n$$
 * For $$A:m\times n \mid m<n$$ we have right inverses; for example, $$A \underbrace{ A^\text{T}\left(AA^\text{T}\right)^{-1} }_{ A^{-1}_\text{right} } = I_m$$

The left inverse can be used to determine the least norm solution of $$Ax = b$$, which is also the least squares formula for regression and is given by $$x = \left(A^\text{T}A\right)^{-1}A^\text{T}b.$$

No rank deficient matrix has any (even one-sided) inverse. However, the Moore–Penrose inverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists.

As an example of matrix inverses, consider:

 * $$A:2 \times 3 =

\begin{bmatrix} 1 & 2 & 3 \\   4 & 5 & 6  \end{bmatrix} $$

So, as m < n, we have a right inverse, $$A^{-1}_\text{right} = A^\text{T} \left(AA^\text{T}\right)^{-1}.$$ By components it is computed as

 * $$\begin{align}

AA^\text{T} &= \begin{bmatrix} 1 & 2 & 3 \\             4 & 5 & 6            \end{bmatrix} \begin{bmatrix} 1 & 4\\             2 & 5\\              3 & 6            \end{bmatrix} = \begin{bmatrix} 14 & 32\\             32 & 77            \end{bmatrix} \\[3pt] \left(AA^\text{T}\right)^{-1} &= \begin{bmatrix} 14 & 32\\      32 & 77 \end{bmatrix}^{-1} = \frac{1}{54} \begin{bmatrix} 77 & -32\\                    -32 & 14                   \end{bmatrix} \\[3pt] A^\text{T}\left(AA^\text{T}\right)^{-1} &= \frac{1}{54} \begin{bmatrix} 1 & 4\\                    2 & 5\\                     3 & 6                   \end{bmatrix} \begin{bmatrix} 77 & -32\\                    -32 & 14                   \end{bmatrix} = \frac{1}{18} \begin{bmatrix} -17 & 8\\                     -2 & 2\\                      13 & -4                   \end{bmatrix} = A^{-1}_\text{right} \end{align}$$

The left inverse doesn't exist, because

A^\text{T}A = \begin{bmatrix} 1 & 4\\            2 & 5\\             3 & 6           \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\            4 & 5 & 6           \end{bmatrix} = \begin{bmatrix} 17 & 22 & 27 \\            22 & 29 & 36\\             27 & 36 & 45           \end{bmatrix} $$

which is a singular matrix, and cannot be inverted.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Inverse element


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Universal algebra

Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.

Basic idea
In universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A.

Arity
An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~x. A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments (also called infix notation), like x ∗ y. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f(x,y,z) or f(x1,...,xn). One way of talking about an algebra, then, is by referring to it as an algebra of a certain type $$\Omega$$, where $$\Omega$$ is an ordered sequence of natural numbers representing the arity of the operations of the algebra. However, some researchers also allow infinitary operations, such as $$\textstyle\bigwedge_{\alpha\in J} x_\alpha$$ where J is an infinite index set, which is an operation in the algebraic theory of complete lattices.

Equations
After the operations have been specified, the nature of the algebra is further defined by axioms, which in universal algebra often take the form of identities, or equational laws.   An example is the associative axiom for a binary operation, which is given by the equation x ∗ (y ∗ z) = (x ∗ y) ∗ z. The axiom is intended to hold for all elements x, y, and z of the set A.

Varieties
A collection of algebraic structures defined by identities is called a variety or equational class.

Restricting one's study to varieties rules out:
 * quantification, including universal quantification (∀) except before an equation, and existential quantification (∃)
 * logical connectives other than conjunction (∧)
 * relations other than equality, in particular inequalities, both a ≠ b and order relations

The study of equational classes can be seen as a special branch of model theory, typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only.

Not all algebraic structures in a wider sense fall into this scope. For example, ordered groups involve an ordering relation, so would not fall within this scope.

The class of fields is not an equational class because there is no type (or "signature") in which all field laws can be written as equations (inverses of elements are defined for all non-zero elements in a field, so inversion cannot be added to the type).

One advantage of this restriction is that the structures studied in universal algebra can be defined in any category that has finite products. For example, a topological group is just a group in the category of topological spaces.

Examples
Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities.

Groups
As an example, consider the definition of a group. Usually a group is defined in terms of a single binary operation ∗, subject to the axioms:
 * Associativity (as in the previous section): x ∗ (y ∗ z)  =  (x ∗ y) ∗ z;   formally: ∀x,y,z. x∗(y∗z)=(x∗y)∗z.
 * Identity element: There exists an element e such that for each element x, one has e ∗ x  =  x  =  x ∗ e;   formally: ∃e ∀x. e∗x=x=x∗e.
 * Inverse element: The identity element is easily seen to be unique, and is usually denoted by e. Then for each x, there exists an element i such that x ∗ i  =  e  =  i ∗ x;   formally: ∀x ∃i. x∗i=e=i∗x.

(Some authors also use the "closure" axiom that x ∗ y belongs to A whenever x and y do, but here this is already implied by calling ∗ a binary operation.)

This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all ..." elements, but also involve the existential quantifier "there exists ...". The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation e and a unary operation ~, with ~x usually written as x−1. The axioms become:
 * Associativity: x ∗ (y ∗ z) = (x ∗ y) ∗ z.
 * Identity element: e ∗ x = x = x ∗ e;   formally: ∀x. e∗x=x=x∗e.
 * Inverse element: x ∗ (~x) = e = (~x) ∗ x;   formally: ∀x. x∗~x=e=~x∗x.

To summarize, the usual definition has: while the universal algebra definition has:
 * a single binary operation (signature (2))
 * 1 equational law (associativity)
 * 2 quantified laws (identity and inverse)
 * 3 operations: one binary, one unary, and one nullary (signature (2, 1, 0))
 * 3 equational laws (associativity, identity, and inverse)
 * no quantified laws (except outermost universal quantifiers, which are allowed in varieties)

A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although the usual definition did not uniquely specify the identity element e, an easy exercise shows that it is unique, as is the inverse of each element.

The universal algebra point of view is well adapted to category theory. For example, when defining a group object in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), rather than quantified laws (which refer to individual elements). Further, the inverse and identity are specified as morphisms in the category. For example, in a topological group, the inverse must not only exist element-wise, but must give a continuous mapping (a morphism). Some authors also require the identity map to be a closed inclusion (a cofibration).

Other examples
Most algebraic structures are examples of universal algebras.
 * Rings, semigroups, quasigroups, groupoids, magmas, loops, and others.
 * Vector spaces over a fixed field and modules over a fixed ring are universal algebras. These have a binary addition and a family of unary scalar multiplication operators, one for each element of the field or ring.

Examples of relational algebras include semilattices, lattices, and Boolean algebras.

Basic constructions
We assume that the type, $$\Omega$$, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.

A homomorphism between two algebras A and B is a function h : A → B from the set A to the set B such that, for every operation fA of A and corresponding fB of B (of arity, say, n), h(fA(x1, ..., xn)) = fB(h(x1), ..., h(xn)). (Sometimes the subscripts on f are taken off when it is clear from context which algebra the function is from.) For example, if e is a constant (nullary operation), then h(eA) = eB. If ~ is a unary operation, then h(~x) = ~h(x). If ∗ is a binary operation, then h(x ∗ y) = h(x) ∗ h(y). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under Homomorphism. In particular, we can take the homomorphic image of an algebra, h(A).

A subalgebra of A is a subset of A that is closed under all the operations of A. A product of some set of algebraic structures is the cartesian product of the sets with the operations defined coordinatewise.

Some basic theorems

 * The isomorphism theorems, which encompass the isomorphism theorems of groups, rings, modules, etc.
 * Birkhoff's HSP Theorem, which states that a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products.

Motivations and applications
In addition to its unifying approach, universal algebra also gives deep theorems and important examples and counterexamples. It provides a useful framework for those who intend to start the study of new classes of algebras. It can enable the use of methods invented for some particular classes of algebras to other classes of algebras, by recasting the methods in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D.H. Smith puts it, "What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."

In particular, universal algebra can be applied to the study of monoids, rings, and lattices. Before universal algebra came along, many theorems (most notably the isomorphism theorems) were proved separately in all of these classes, but with universal algebra, they can be proven once and for all for every kind of algebraic system.

The 1956 paper by Higgins referenced below has been well followed up for its framework for a range of particular algebraic systems, while his 1963 paper is notable for its discussion of algebras with operations which are only partially defined, typical examples for this being categories and groupoids. This leads on to the subject of higher-dimensional algebra which can be defined as the study of algebraic theories with partial operations whose domains are defined under geometric conditions. Notable examples of these are various forms of higher-dimensional categories and groupoids.

Constraint satisfaction problem
Universal algebra provides a natural language for the constraint satisfaction problem (CSP). CSP refers to an important class of computational problems where, given a relational algebra A and an existential sentence $$\varphi$$ over this algebra, the question is to find out whether $$\varphi$$ can be satisfied in A. The algebra A is often fixed, so that CSPA refers to the problem whose instance is only the existential sentence $$\varphi$$.

It is proved that every computational problem can be formulated as CSPA for some algebra A.

For example, the n-coloring problem can be stated as CSP of the algebra ($\{0, 1, ..., n−1\}$, ≠), i.e. an algebra with n elements and a single relation, inequality.

The dichotomy conjecture (proved in April 2017) states that if A is a finite algebra, then CSPA is either P or NP-complete.

Generalizations
Universal algebra has also been studied using the techniques of category theory. In this approach, instead of writing a list of operations and equations obeyed by those operations, one can describe an algebraic structure using categories of a special sort, known as Lawvere theories or more generally algebraic theories. Alternatively, one can describe algebraic structures using monads. The two approaches are closely related, with each having their own advantages. In particular, every Lawvere theory gives a monad on the category of sets, while any "finitary" monad on the category of sets arises from a Lawvere theory. However, a monad describes algebraic structures within one particular category (for example the category of sets), while algebraic theories describe structure within any of a large class of categories (namely those having finite products).

A more recent development in category theory is operad theory – an operad is a set of operations, similar to a universal algebra, but restricted in that equations are only allowed between expressions with the variables, with no duplication or omission of variables allowed. Thus, rings can be described as the so-called "algebras" of some operad, but not groups, since the law gg−1 = 1 duplicates the variable g on the left side and omits it on the right side. At first this may seem to be a troublesome restriction, but the payoff is that operads have certain advantages: for example, one can hybridize the concepts of ring and vector space to obtain the concept of associative algebra, but one cannot form a similar hybrid of the concepts of group and vector space.

Another development is partial algebra where the operators can be partial functions. Certain partial functions can also be handled by a generalization of Lawvere theories known as "essentially algebraic theories".

Another generalization of universal algebra is model theory, which is sometimes described as "universal algebra + logic".

History
In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra had essentially the same meaning that it has today. Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself.

At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander Macfarlane wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures." At the time George Boole's algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.

Whitehead's early work sought to unify quaternions (due to Hamilton), Grassmann's Ausdehnungslehre, and Boole's algebra of logic. Whitehead wrote in his book:
 * "Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge."

Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when Garrett Birkhoff and Øystein Ore began publishing on universal algebras. Developments in metamathematics and category theory in the 1940s and 1950s furthered the field, particularly the work of Abraham Robinson, Alfred Tarski, Andrzej Mostowski, and their students.

In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoff's papers, dealing with free algebras, congruence and subalgebra lattices, and homomorphism theorems. Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by Anatoly Maltsev in the 1940s went unnoticed because of the war. Tarski's lecture at the 1950 International Congress of Mathematicians in Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang, Leon Henkin, Bjarni Jónsson, Roger Lyndon, and others.

In the late 1950s, Edward Marczewski emphasized the importance of free algebras, leading to the publication of more than 50 papers on the algebraic theory of free algebras by Marczewski himself, together with Jan Mycielski, Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski, K. Urbanik, and others.

Starting with William Lawvere's thesis in 1963, techniques from category theory have become important in universal algebra.

title: Universal algebra


text:
371

Year 371 (CCCLXXI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Augustus and Petronius (or, less frequently, year 1124 Ab urbe condita). The denomination 371 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * The fortified cities of the Danube, with Sirmium (Pannonia) at the forefront, contribute to stop an invasion of the Quadi.

Persia

 * The neo-Persian Empire attains the zenith of its power under King Shapur II, as the Romans renew their war against Persia. Hostilities will continue for the next 5 years.

Asia

 * Baekje forces storm the Goguryeo capital in P'yongyang (Korea).
 * Sosurim becomes king of Goguryeo.

Art and Science

 * Roman poet Ausonius writes of a voyage on the Rhine and the Moselle, in his work 'Mosella'.

Religion

 * Augustine of Hippo, age 17, travels to Carthage, to continue his education in rhetoric.
 * Martin of Tours becomes bishop of Tours (approximate date).

Births

 * Dao Wu Di, Chinese emperor of Northern Wei (d. 409)
 * Sengrui, Chinese Buddhist monk and scholar (d. 438)
 * Valentinian II, Roman consul and emperor (d. 392)

Deaths

 * April 12 – Zeno of Verona, Christian bishop and martyr
 * August 1 – Eusebius of Vercelli, Christian bishop (b. 283)
 * Gogugwon, king of Goguryeo (Korea)
 * Hilarion, Syrian anchorite and saint (b. 291)
 * Lucifer Calaritanus, founder of the Luciferian sect (approximate date)

title: 371


text:
372

Year 372 (CCCLXXII) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Modestus and Arintheus (or, less frequently, year 1125 Ab urbe condita). The denomination 372 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * Emperor Valentinian I is engaged in operations against the Alamanni, Quadi and Sarmatians, while his subordinates are dealing with Firmus,  Roman usurper, in Africa and the Picts in Britain.

Europe

 * The Huns attack the Tervingi on the Dniester, overwhelming them with light cavalry (horse archers), and devastating the settlements of the Goths. King Athanaric is defeated, and seeks refuge in the Carpathian Mountains (Romania).
 * Athanaric starts building new defensive works, to protect his people against the Alans and the Huns.

China

 * Sixteen Kingdoms: Jin Feidi is dethroned as emperor of the Eastern Jin Dynasty. He is replaced by his granduncle Sima Yu, who is installed as Jin Jianwendi.
 * September 12 – Jin Xiaowudi, age 10, succeeds his father Jin Jianwendi. Empress Chu Suanzi serves as regent, but decisions are made by the high officials Xie An and Wang Tanzhi.
 * The first diplomatic ties are established between the Korean kingdom of Baekje and the Chinese court of the Jin Dynasty.

Art and Science

 * The national academy of Chinese learning, called Taehak, is established in the kingdom of Goguryeo (Korea).

Religion

 * Gregory of Nyssa becomes bishop.
 * Buddhism is adopted as the official religion of Gorguryeo.
 * Saint Augustine adopts Manichaeism.
 * Valentinian I bans Manichaean meetings.

Births

 * Pei Songzhi, Chinese historian and politician (d. 451)

Deaths

 * Jianwen of Jin, Chinese emperor of the Jin Dynasty (b. 320)
 * Maximus of Ephesus, Greek Neoplatonist philosopher
 * Sabbas the Goth, Christian reader and saint (b. 334)

title: 372


text:
Submarine communications cable

Submarine cable cross-section 3D plain.svg of the shore-end of a modern submarine communications cable.

1 – Polyethylene

2 – Mylar tape

3 – Stranded steel wires

4 – Aluminium water barrier

5 – Polycarbonate

6 – Copper or aluminium tube

7 – Petroleum jelly

8 – Optical fibers]]

A submarine communications cable is a cable laid on the seabed between land-based stations to carry telecommunication signals across stretches of ocean and sea. The first submarine communications cables were laid beginning in the 1850s and carried telegraphy traffic, establishing the first instant telecommunications links between continents, such as the first transatlantic telegraph cable which became operational on 16 August 1858.

Submarine cables first connected all the world's continents (except Antarctica) when Java was connected to Darwin, Northern Territory, Australia, in 1871 in anticipation of the completion of the Australian Overland Telegraph Line in 1872 connecting to Adelaide, South Australia and thence to the rest of Australia.

Subsequent generations of cables carried telephone traffic, then data communications traffic. These early cables used copper wires in their cores, but modern cables use optical fiber technology to carry digital data, which includes telephone, Internet and private data traffic. Modern cables are typically about 25 mm in diameter and weigh around 1.4 t/km for the deep-sea sections which comprise the majority of the run, although larger and heavier cables are used for shallow-water sections near shore.

First successful trials
After William Cooke and Charles Wheatstone had introduced their working telegraph in 1839, the idea of a submarine line across the Atlantic Ocean began to be thought of as a possible triumph of the future. Samuel Morse proclaimed his faith in it as early as 1840, and in 1842, he submerged a wire, insulated with tarred hemp and India rubber, in the water of New York Harbor, and telegraphed through it. The following autumn, Wheatstone performed a similar experiment in Swansea Bay. A good insulator to cover the wire and prevent the electric current from leaking into the water was necessary for the success of a long submarine line. India rubber had been tried by Moritz von Jacobi, the Prussian electrical engineer, as far back as the early 19th century.

Another insulating gum which could be melted by heat and readily applied to wire made its appearance in 1842. Gutta-percha, the adhesive juice of the Palaquium gutta tree, was introduced to Europe by William Montgomerie, a Scottish surgeon in the service of the British East India Company. Twenty years earlier, Montgomerie had seen whips made of gutta-percha in Singapore, and he believed that it would be useful in the fabrication of surgical apparatus. Michael Faraday and Wheatstone soon discovered the merits of gutta-percha as an insulator, and in 1845, the latter suggested that it should be employed to cover the wire which was proposed to be laid from Dover to Calais. In 1847 William Siemens, then an officer in the army of Prussia, laid the first successful underwater cable using gutta percha insulation, across the Rhine between Deutz and Cologne. In 1849, Charles Vincent Walker, electrician to the South Eastern Railway, submerged 2 mi of wire coated with gutta-percha off the coast from Folkestone, which was tested successfully.

First commercial cables

In August 1850, having earlier obtained a concession from the French government, John Watkins Brett's English Channel Submarine Telegraph Company laid the first line across the English Channel, using the converted tugboat Goliath. It was simply a copper wire coated with gutta-percha, without any other protection, and was not successful. However, the experiment served to secure renewal of the concession, and in September 1851, a protected core, or true, cable was laid by the reconstituted Submarine Telegraph Company from a government hulk, Blazer, which was towed across the Channel.

In 1853, more successful cables were laid, linking Great Britain with Ireland, Belgium, and the Netherlands, and crossing The Belts in Denmark. The British & Irish Magnetic Telegraph Company completed the first successful Irish link on May 23 between Portpatrick and Donaghadee using the collier William Hutt. The same ship was used for the link from Dover to Ostend in Belgium, by the Submarine Telegraph Company. Meanwhile, the Electric & International Telegraph Company completed two cables across the North Sea, from Orford Ness to Scheveningen, the Netherlands. These cables were laid by Monarch, a paddle steamer which later became the first vessel with permanent cable-laying equipment.

In 1858, the steamship Elba was used to lay a telegraph cable from Jersey to Guernsey, on to Alderney and then to Weymouth, the cable being completed successfully in September of that year. Problems soon developed with eleven breaks occurring by 1860 due to storms, tidal and sand movements, and wear on rocks. A report to the Institution of Civil Engineers in 1860 set out the problems to assist in future cable-laying operations.

Crimean War (1853–1856)
In the Crimean War various forms of telegraphy played a major role; this was a first. At the start of the campaign there was a telegraph link at Bucharest connected to London. In the winter of 1854 the French extended the telegraph link to the Black Sea coast. In April 1855 the British laid an underwater cable from Varna to the Crimean peninsula so that news of the Crimean War could reach London in a handful of hours.

Transatlantic telegraph cable
The first attempt at laying a transatlantic telegraph cable was promoted by Cyrus West Field, who persuaded British industrialists to fund and lay one in 1858. However, the technology of the day was not capable of supporting the project; it was plagued with problems from the outset, and was in operation for only a month. Subsequent attempts in 1865 and 1866 with the world's largest steamship, the SS Great Eastern, used a more advanced technology and produced the first successful transatlantic cable. Great Eastern later went on to lay the first cable reaching to India from Aden, Yemen, in 1870.

British dominance of early cable

From the 1850s until 1911, British submarine cable systems dominated the most important market, the North Atlantic Ocean. The British had both supply side and demand side advantages. In terms of supply, Britain had entrepreneurs willing to put forth enormous amounts of capital necessary to build, lay and maintain these cables. In terms of demand, Britain's vast colonial empire led to business for the cable companies from news agencies, trading and shipping companies, and the British government. Many of Britain's colonies had significant populations of European settlers, making news about them of interest to the general public in the home country.

British officials believed that depending on telegraph lines that passed through non-British territory posed a security risk, as lines could be cut and messages could be interrupted during wartime. They sought the creation of a worldwide network within the empire, which became known as the All Red Line, and conversely prepared strategies to quickly interrupt enemy communications. Britain's very first action after declaring war on Germany in World War I was to have the cable ship Alert (not the CS Telconia as frequently reported) cut the five cables linking Germany with France, Spain and the Azores, and through them, North America. Thereafter, the only way Germany could communicate was by wireless, and that meant that Room 40 could listen in.

The submarine cables were an economic benefit to trading companies, because owners of ships could communicate with captains when they reached their destination and give directions as to where to go next to pick up cargo based on reported pricing and supply information. The British government had obvious uses for the cables in maintaining administrative communications with governors throughout its empire, as well as in engaging other nations diplomatically and communicating with its military units in wartime. The geographic location of British territory was also an advantage as it included both Ireland on the east side of the Atlantic Ocean and Newfoundland in North America on the west side, making for the shortest route across the ocean, which reduced costs significantly.

A few facts put this dominance of the industry in perspective. In 1896, there were 30 cable-laying ships in the world, 24 of which were owned by British companies. In 1892, British companies owned and operated two-thirds of the world's cables and by 1923, their share was still 42.7 percent. During World War I, Britain's telegraph communications were almost completely uninterrupted, while it was able to quickly cut Germany's cables worldwide.

Cable to India, Singapore, East Asia and Australia

Throughout the 1860s and 1870s, British cable expanded eastward, into the Mediterranean Sea and the Indian Ocean. An 1863 cable to Bombay (now Mumbai), India, provided a crucial link to Saudi Arabia. In 1870, Bombay was linked to London via submarine cable in a combined operation by four cable companies, at the behest of the British Government. In 1872, these four companies were combined to form the mammoth globe-spanning Eastern Telegraph Company, owned by John Pender. A spin-off from Eastern Telegraph Company was a second sister company, the Eastern Extension, China and Australasia Telegraph Company, commonly known simply as "the Extension." In 1872, Australia was linked by cable to Bombay via Singapore and China and in 1876, the cable linked the British Empire from London to New Zealand.

Submarine cables across the Pacific
The first trans-Pacific cables providing telegraph service were completed in 1902 and 1903, linking the US mainland to Hawaii in 1902 and Guam to the Philippines in 1903. Canada, Australia, New Zealand and Fiji were also linked in 1902 with the trans-Pacific segment of the All Red Line. Japan was connected into the system in 1906. Service beyond Midway Atoll was abandoned in 1941 due to World War II, but the remainder stayed in operation until 1951 when the FCC gave permission to cease operations.

The first trans-Pacific telephone cable was laid from Hawaii to Japan in 1964, with an extension from Guam to The Philippines. Also in 1964, the Commonwealth Pacific Cable System (COMPAC), with 80 telephone channel capacity, opened for traffic from Sydney to Vancouver, and in 1967, the South East Asia Commonwealth (SEACOM) system, with 160 telephone channel capacity, opened for traffic. This system used microwave radio from Sydney to Cairns (Queensland), cable running from Cairns to Madang (Papua New Guinea), Guam, Hong Kong, Kota Kinabalu (capital of Sabah, Malaysia), Singapore, then overland by microwave radio to Kuala Lumpur. In 1991, the North Pacific Cable system was the first regenerative system (i.e., with repeaters) to completely cross the Pacific from the US mainland to Japan. The US portion of NPC was manufactured in Portland, Oregon, from 1989 to 1991 at STC Submarine Systems, and later Alcatel Submarine Networks. The system was laid by Cable & Wireless Marine on the CS Cable Venture.

Construction
Transatlantic cables of the 19th century consisted of an outer layer of iron and later steel wire, wrapping India rubber, wrapping gutta-percha, which surrounded a multi-stranded copper wire at the core. The portions closest to each shore landing had additional protective armour wires. Gutta-percha, a natural polymer similar to rubber, had nearly ideal properties for insulating submarine cables, with the exception of a rather high dielectric constant which made cable capacitance high. William Thomas Henley had developed a machine in 1837 for covering wires with silk or cotton thread that he developed into a wire wrapping capability for submarine cable with a factory in 1857 that became W.T. Henley's Telegraph Works Co., Ltd. The India Rubber, Gutta Percha and Telegraph Works Company, established by the Silver family and giving that name to a section of London, furnished cores to Henley's as well as eventually making and laying finished cable. In 1870 William Hooper established Hooper's Telegraph Works to manufacture his patented vulcanized rubber core, at first to furnish other makers of finished cable, that began to compete with the gutta-percha cores. The company later expanded into complete cable manufacture and cable laying, including the building of the first cable ship specifically designed to lay transatlantic cables.

Gutta-percha and rubber were not replaced as a cable insulation until polyethylene was introduced in the 1930s. Even then, the material was only available to the military and the first submarine cable using it was not laid until 1945 during World War II across the English Channel. In the 1920s, the American military experimented with rubber-insulated cables as an alternative to gutta-percha, since American interests controlled significant supplies of rubber but did not have easy access to gutta-percha manufacturers. The 1926 development by John T. Blake of deproteinized rubber improved the impermeability of cables to water.

Many early cables suffered from attack by sea life. The insulation could be eaten, for instance, by species of Teredo (shipworm) and Xylophaga. Hemp laid between the steel wire armouring gave pests a route to eat their way in. Damaged armouring, which was not uncommon, also provided an entrance. Cases of sharks biting cables and attacks by sawfish have been recorded. In one case in 1873, a whale damaged the Persian Gulf Cable between Karachi and Gwadar. The whale was apparently attempting to use the cable to clean off barnacles at a point where the cable descended over a steep drop. The unfortunate whale got its tail entangled in loops of cable and drowned. The cable repair ship Amber Witch was only able to winch up the cable with difficulty, weighed down as it was with the dead whale's body.

Bandwidth problems
Early long-distance submarine telegraph cables exhibited formidable electrical problems. Unlike modern cables, the technology of the 19th century did not allow for in-line repeater amplifiers in the cable. Large voltages were used to attempt to overcome the electrical resistance of their tremendous length but the cables' distributed capacitance and inductance combined to distort the telegraph pulses in the line, reducing the cable's bandwidth, severely limiting the data rate for telegraph operation to 10–12 words per minute.

As early as 1816, Francis Ronalds had observed that electric signals were slowed in passing through an insulated wire or core laid underground, and outlined the cause to be induction, using the analogy of a long Leyden jar. The same effect was noticed by Latimer Clark (1853) on cores immersed in water, and particularly on the lengthy cable between England and The Hague. Michael Faraday showed that the effect was caused by capacitance between the wire and the earth (or water) surrounding it. Faraday had noticed that when a wire is charged from a battery (for example when pressing a telegraph key), the electric charge in the wire induces an opposite charge in the water as it travels along. In 1831, Faraday described this effect in what is now referred to as Faraday's law of induction. As the two charges attract each other, the exciting charge is retarded. The core acts as a capacitor distributed along the length of the cable which, coupled with the resistance and inductance of the cable, limits the speed at which a signal travels through the conductor of the cable.

Early cable designs failed to analyse these effects correctly. Famously, E.O.W. Whitehouse had dismissed the problems and insisted that a transatlantic cable was feasible. When he subsequently became chief electrician of the Atlantic Telegraph Company, he became involved in a public dispute with William Thomson. Whitehouse believed that, with enough voltage, any cable could be driven. Thomson believed that his law of squares showed that retardation could not be overcome by a higher voltage. His recommendation was a larger cable. Because of the excessive voltages recommended by Whitehouse, Cyrus West Field's first transatlantic cable never worked reliably, and eventually short circuited to the ocean when Whitehouse increased the voltage beyond the cable design limit.

Thomson designed a complex electric-field generator that minimized current by resonating the cable, and a sensitive light-beam mirror galvanometer for detecting the faint telegraph signals. Thomson became wealthy on the royalties of these, and several related inventions. Thomson was elevated to Lord Kelvin for his contributions in this area, chiefly an accurate mathematical model of the cable, which permitted design of the equipment for accurate telegraphy. The effects of atmospheric electricity and the geomagnetic field on submarine cables also motivated many of the early polar expeditions.

Thomson had produced a mathematical analysis of propagation of electrical signals into telegraph cables based on their capacitance and resistance, but since long submarine cables operated at slow rates, he did not include the effects of inductance. By the 1890s, Oliver Heaviside had produced the modern general form of the telegrapher's equations, which included the effects of inductance and which were essential to extending the theory of transmission lines to the higher frequencies required for high-speed data and voice.

Transatlantic telephony

While laying a transatlantic telephone cable was seriously considered from the 1920s, the technology required for economically feasible telecommunications was not developed until the 1940s. A first attempt to lay a "pupinized" telephone cable—one with loading coils added at regular intervals—failed in the early 1930s due to the Great Depression.

TAT-1 (Transatlantic No. 1) was the first transatlantic telephone cable system. Between 1955 and 1956, cable was laid between Gallanach Bay, near Oban, Scotland and Clarenville, Newfoundland and Labrador, in Canada. It was inaugurated on September 25, 1956, initially carrying 36 telephone channels.

In the 1960s, transoceanic cables were coaxial cables that transmitted frequency-multiplexed voiceband signals. A high-voltage direct current on the inner conductor powered repeaters (two-way amplifiers placed at intervals along the cable). The first-generation repeaters remain among the most reliable vacuum tube amplifiers ever designed. Later ones were transistorized. Many of these cables are still usable, but have been abandoned because their capacity is too small to be commercially viable. Some have been used as scientific instruments to measure earthquake waves and other geomagnetic events.

Other uses
In 1942, Siemens Brothers of New Charlton, London, in conjunction with the United Kingdom National Physical Laboratory, adapted submarine communications cable technology to create the world's first submarine oil pipeline in Operation Pluto during World War II.

Active fiber-optic cables may be useful in detecting seismic events which alter cable polarization.

Optical telecommunications cables

In the 1980s, fiber-optic cables were developed. The first transatlantic telephone cable to use optical fiber was TAT-8, which went into operation in 1988. A fiber-optic cable comprises multiple pairs of fibers. Each pair has one fiber in each direction. TAT-8 had two operational pairs and one backup pair. Except for very short lines, fiber-optic submarine cables include repeaters at regular intervals.

Modern optical fiber repeaters use a solid-state optical amplifier, usually an erbium-doped fiber amplifier (EDFA). Each repeater contains separate equipment for each fiber. These comprise signal reforming, error measurement and controls. A solid-state laser dispatches the signal into the next length of fiber. The solid-state laser excites a short length of doped fiber that itself acts as a laser amplifier. As the light passes through the fiber, it is amplified. This system also permits wavelength-division multiplexing, which dramatically increases the capacity of the fiber. EDFA amplifiers were first used in submarine cables in 1995.

Repeaters are powered by a constant direct current passed down the conductor near the centre of the cable, so all repeaters in a cable are in series. Power feed equipment is installed at the terminal stations. Typically both ends share the current generation with one end providing a positive voltage and the other a negative voltage. A virtual earth point exists roughly halfway along the cable under normal operation. The amplifiers or repeaters derive their power from the potential difference across them. The voltage passed down the cable is often anywhere from 3000 to 15,000VDC at a current of up to 1,100mA, with the current increasing with decreasing voltage; the current at 10,000VDC is up to 1,650mA. Hence the total amount of power sent into the cable is often up to 16.5 kW.

The optic fiber used in undersea cables is chosen for its exceptional clarity, permitting runs of more than 100 km between repeaters to minimize the number of amplifiers and the distortion they cause. Unrepeated cables are cheaper than repeated cables and their maximum transmission distance is limited, although this has increased over the years; in 2014 unrepeated cables of up to 380 km in length were in service; however these require unpowered repeaters to be positioned every 100 km.

The rising demand for these fiber-optic cables outpaced the capacity of providers such as AT&T. Having to shift traffic to satellites resulted in lower-quality signals. To address this issue, AT&T had to improve its cable-laying abilities. It invested $100 million in producing two specialized fiber-optic cable laying vessels. These included laboratories in the ships for splicing cable and testing its electrical properties. Such field monitoring is important because the glass of fiber-optic cable is less malleable than the copper cable that had been formerly used. The ships are equipped with thrusters that increase maneuverability. This capability is important because fiber-optic cable must be laid straight from the stern, which was another factor that copper-cable-laying ships did not have to contend with.

Originally, submarine cables were simple point-to-point connections. With the development of submarine branching units (SBUs), more than one destination could be served by a single cable system. Modern cable systems now usually have their fibers arranged in a self-healing ring to increase their redundancy, with the submarine sections following different paths on the ocean floor. One reason for this development was that the capacity of cable systems had become so large that it was not possible to completely back up a cable system with satellite capacity, so it became necessary to provide sufficient terrestrial backup capability. Not all telecommunications organizations wish to take advantage of this capability, so modern cable systems may have dual landing points in some countries (where back-up capability is required) and only single landing points in other countries where back-up capability is either not required, the capacity to the country is small enough to be backed up by other means, or having backup is regarded as too expensive.

A further redundant-path development over and above the self-healing rings approach is the mesh network whereby fast switching equipment is used to transfer services between network paths with little to no effect on higher-level protocols if a path becomes inoperable. As more paths become available to use between two points, it is less likely that one or two simultaneous failures will prevent end-to-end service.

As of 2012, operators had "successfully demonstrated long-term, error-free transmission at 100 Gbps across Atlantic Ocean" routes of up to 6000 km, meaning a typical cable can move tens of terabits per second overseas. Speeds improved rapidly in the previous few years, with 40 Gbit/s having been offered on that route only three years earlier in August 2009.

Switching and all-by-sea routing commonly increases the distance and thus the round trip latency by more than 50%. For example, the round trip delay (RTD) or latency of the fastest transatlantic connections is under 60 ms, close to the theoretical optimum for an all-sea route. While in theory, a great circle route (GCP) between London and New York City is only 5600 km, this requires several land masses (Ireland, Newfoundland, Prince Edward Island and the isthmus connecting New Brunswick to Nova Scotia) to be traversed, as well as the extremely tidal Bay of Fundy and a land route along Massachusetts' north shore from Gloucester to Boston and through fairly built up areas to Manhattan itself. In theory, using this partial land route could result in round trip times below 40 ms (which is the speed of light minimum time), and not counting switching. Along routes with less land in the way, round trip times can approach speed of light minimums in the long term.

The type of optical fiber used in unrepeated and very long cables is often PCSF (pure silica core) due to its low loss of 0.172 dB per kilometer when carrying a 1550 nm wavelength laser light. The large chromatic dispersion of PCSF means that its use requires transmission and receiving equipment designed with this in mind; this property can also be used to reduce interference when transmitting multiple channels through a single fiber using wavelength division multiplexing (WDM), which allows for multiple optical carrier channels to be transmitted through a single fiber, each carrying its own information. WDM is limited by the optical bandwidth of the amplifiers used to transmit data through the cable and by the spacing between the frequencies of the optical carriers; however this minimum spacing is also limited, with the minimum spacing often being 50 GHz (0.4 nm). The use of WDM can reduce the maximum length of the cable although this can be overcome by designing equipment with this in mind.

Optical post amplifiers, used to increase the strength of the signal generated by the optical transmitter often use a diode-pumped erbium-doped fiber laser. The diode is often a high power 980 or 1480 nm laser diode. This setup allows for an amplification of up to +24dBm in an affordable manner. Using an erbium-ytterbium doped fiber instead allows for a gain of +33dBm, however again the amount of power that can be fed into the fiber is limited. In single carrier configurations the dominating limitation is self phase modulation induced by the Kerr effect which limits the amplification to +18 dBm per fiber. In WDM configurations the limitation due to crossphase modulation becomes predominant instead. Optical pre-amplifiers are often used to negate the thermal noise of the receiver. Pumping the pre-amplifier with a 980 nm laser leads to a noise of at most 3.5 dB, with a noise of 5 dB usually obtained with a 1480 nm laser. The noise has to be filtered using optical filters.

Raman amplification can be used to extend the reach or the capacity of an unrepeatered cable, by launching 2 frequencies into a single fiber; one carrying data signals at 1550 nm, and the other pumping them at 1450 nm. Launching a pump frequency (pump laser light) at a power of just one watt leads to an increase in reach of 45 km or a 6-fold increase in capacity.

Another way to increase the reach of a cable is by using unpowered repeaters called remote optical pre-amplifiers (ROPAs); these still make a cable count as unrepeatered since the repeaters do not require electrical power but they do require a pump laser light to be transmitted alongside the data carried by the cable; the pump light and the data are often transmitted in physically separate fibers. The ROPA contains a doped fiber that uses the pump light (often a 1480 nm laser light) to amplify the data signals carried on the rest of the fibers.

WDM or wavelength division multiplexing was first implemented in submarine fiber optic cables from the 1990s to the 2000s, followed by DWDM or dense wavelength division mulltiplexing around 2007. Each fiber can carry 30 wavelengths at a time. SDM or spatial division multiplexing submarine cables have at least 12 fiber pairs which is an increase from the maximum of 8 pairs found in conventional submarine cables, and submarine cables with up to 24 fiber pairs have been deployed. The type of modulation employed in a submarine cable can have a major impact in its capacity. SDM is combined with DWDM to improve capacity.

The open cable concept allows for the design of a submarine cable independently of the transponders that will be used to transmit data through the cable. SLTE (Submarine Line Terminal Equipment) has transponders and a ROADM (Reconfigurable optical add-drop multiplexer) used for handling the signals in the cable via software control. The ROADM is used to improve the reliability of the cable by allowing it to operate even if it has faults. This equipment is located inside a cable landing station (CLS). C-OTDR (Coherent Optical Time Domain Reflectometry) is used in submarine cables to detect the location of cable faults. The wet plant of a submarine cable comprises the cable itself, branching units, repeaters and possibly OADMs (Optical add-drop multiplexers).

Importance of submarine cables
Currently 99% of the data traffic that is crossing oceans is carried by undersea cables. The reliability of submarine cables is high, especially when (as noted above) multiple paths are available in the event of a cable break. Also, the total carrying capacity of submarine cables is in the terabits per second, while satellites typically offer only 1,000 megabits per second and display higher latency. However, a typical multi-terabit, transoceanic submarine cable system costs several hundred million dollars to construct.

As a result of these cables' cost and usefulness, they are highly valued not only by the corporations building and operating them for profit, but also by national governments. For instance, the Australian government considers its submarine cable systems to be "vital to the national economy". Accordingly, the Australian Communications and Media Authority (ACMA) has created protection zones that restrict activities that could potentially damage cables linking Australia to the rest of the world. The ACMA also regulates all projects to install new submarine cables.

Submarine cables are important to the modern military as well as private enterprise. The US military, for example, uses the submarine cable network for data transfer from conflict zones to command staff in the United States. Interruption of the cable network during intense operations could have direct consequences for the military on the ground.

Investment and finances

Almost all fiber-optic cables from TAT-8 in 1988 until approximately 1997 were constructed by consortia of operators. For example, TAT-8 counted 35 participants including most major international carriers at the time such as AT&T Corporation. Two privately financed, non-consortium cables were constructed in the late 1990s, which preceded a massive, speculative rush to construct privately financed cables that peaked in more than $22 billion worth of investment between 1999 and 2001. This was followed by the bankruptcy and reorganization of cable operators such as Global Crossing, 360networks, FLAG, Worldcom, and Asia Global Crossing. Tata Communications' Global Network (TGN) is the only wholly owned fiber network circling the planet.

Most cables in the 20th century crossed the Atlantic Ocean, to connect the United States and Europe. However, capacity in the Pacific Ocean was much expanded starting in the 1990s. For example, between 1998 and 2003, approximately 70% of undersea fiber-optic cable was laid in the Pacific. This is in part a response to the emerging significance of Asian markets in the global economy.

After decades of heavy investment in already developed markets such as the transatlantic and transpacific routes, efforts increased in the 21st century to expand the submarine cable network to serve the developing world. For instance, in July 2009, an underwater fiber-optic cable line plugged East Africa into the broader Internet. The company that provided this new cable was SEACOM, which is 75% owned by East African and South African investors. The project was delayed by a month due to increased piracy along the coast.

Investments in cables present a commercial risk because cables cover 6,200 km of ocean floor, cross submarine mountain ranges and rifts. Because of this most companies only purchase capacity after the cable is finished.

Antarctica
Antarctica is the only continent not yet reached by a submarine telecommunications cable. Phone, video, and e-mail traffic must be relayed to the rest of the world via satellite links that have limited availability and capacity. Bases on the continent itself are able to communicate with one another via radio, but this is only a local network. To be a viable alternative, a fiber-optic cable would have to be able to withstand temperatures of -80 C as well as massive strain from ice flowing up to 10 m per year. Thus, plugging into the larger Internet backbone with the high bandwidth afforded by fiber-optic cable is still an as-yet infeasible economic and technical challenge in the Antarctic.

Arctic
Several projects are underway in the Arctic including 12,650 km "Polar Express" and 14,500 km Far North Fiber.

Cable repair

Cables can be broken by fishing trawlers, anchors, earthquakes, turbidity currents, and even shark bites. Based on surveying breaks in the Atlantic Ocean and the Caribbean Sea, it was found that between 1959 and 1996, fewer than 9% were due to natural events. In response to this threat to the communications network, the practice of cable burial has developed. The average incidence of cable faults was 3.7 per 1000 km per year from 1959 to 1979. That rate was reduced to 0.44 faults per 1,000 km per year after 1985, due to widespread burial of cable starting in 1980. Still, cable breaks are by no means a thing of the past, with more than 50 repairs a year in the Atlantic alone, and significant breaks in 2006, 2008, 2009 and 2011.

The propensity for fishing trawler nets to cause cable faults may well have been exploited during the Cold War. For example, in February 1959, a series of 12 breaks occurred in five American trans-Atlantic communications cables. In response, a United States naval vessel, the USS Roy O. Hale, detained and investigated the Soviet trawler Novorosiysk. A review of the ship's log indicated it had been in the region of each of the cables when they broke. Broken sections of cable were also found on the deck of the Novorosiysk. It appeared that the cables had been dragged along by the ship's nets, and then cut once they were pulled up onto the deck to release the nets. The Soviet Union's stance on the investigation was that it was unjustified, but the United States cited the Convention for the Protection of Submarine Telegraph Cables of 1884 to which Russia had signed (prior to the formation of the Soviet Union) as evidence of violation of international protocol.

Shore stations can locate a break in a cable by electrical measurements, such as through spread-spectrum time-domain reflectometry (SSTDR), a type of time-domain reflectometry that can be used in live environments very quickly. Presently, SSTDR can collect a complete data set in 20 ms. Spread spectrum signals are sent down the wire and then the reflected signal is observed. It is then correlated with the copy of the sent signal and algorithms are applied to the shape and timing of the signals to locate the break.

A cable repair ship will be sent to the location to drop a marker buoy near the break. Several types of grapples are used depending on the situation. If the sea bed in question is sandy, a grapple with rigid prongs is used to plough under the surface and catch the cable. If the cable is on a rocky sea surface, the grapple is more flexible, with hooks along its length so that it can adjust to the changing surface. In especially deep water, the cable may not be strong enough to lift as a single unit, so a special grapple that cuts the cable soon after it has been hooked is used and only one length of cable is brought to the surface at a time, whereupon a new section is spliced in. The repaired cable is longer than the original, so the excess is deliberately laid in a "U" shape on the seabed. A submersible can be used to repair cables that lie in shallower waters.

A number of ports near important cable routes became homes to specialized cable repair ships. Halifax, Nova Scotia, was home to a half dozen such vessels for most of the 20th century including long-lived vessels such as the CS Cyrus West Field, CS Minia and CS Mackay-Bennett. The latter two were contracted to recover victims from the sinking of the RMS Titanic. The crews of these vessels developed many new techniques and devices to repair and improve cable laying, such as the "plough".

Intelligence gathering
Underwater cables, which cannot be kept under constant surveillance, have tempted intelligence-gathering organizations since the late 19th century. Frequently at the beginning of wars, nations have cut the cables of the other sides to redirect the information flow into cables that were being monitored. The most ambitious efforts occurred in World War I, when British and German forces systematically attempted to destroy the others' worldwide communications systems by cutting their cables with surface ships or submarines. During the Cold War, the United States Navy and National Security Agency (NSA) succeeded in placing wire taps on Soviet underwater communication lines in Operation Ivy Bells. In modern times, the widespread use of end-to-end encryption minimizes the threat of wire tapping.

Environmental impact
The presence of cables in the oceans can be a danger to marine life. With the proliferation of cable installations and the increasing demand for inter-connectivity that today's society demands, the environmental impact is increasing.

Submarine cables can impact marine life in a number of ways.

Alteration of the seabed
Seabed ecosystems can be disturbed by the installation and maintenance of cables. The effects of cable installation are generally limited to specific areas. The intensity of disturbance depends on the installation method.

Cables are often laid in the so-called benthic zone of the seabed. The benthic zone is the ecological region at the bottom of the sea where benthos, clams and crabs live, and where the surface sediments, which are deposits of matter and particles in the water that provide a habitat for marine species, are located.

Sediment can be damaged by cable installation by trenching with water jets or ploughing. This can lead to reworking of the sediments, altering the substrate of which they are composed.

According to several studies, the biota of the benthic zone is only slightly affected by the presence of cables. However, the presence of cables can trigger behavioral disturbances in living organisms. The main observation is that the presence of cables provides a hard substrate for anemones attachment. These organisms are found in large number around cables that run through soft sediments, which are not normally suitable for these organisms. This is also the case for flatfish. Although little observed, the presence of cables can also change the water temperature and therefore disturb the surrounding natural habitat.

However, these disturbances are not very persistent over time, and can stabilize within a few days. Cable operators are trying to implement measures to route cables in such a way as to avoid areas with sensitive and vulnerable ecosystems.

Entanglement
Entanglement of marine animals in cables is one of the main causes of cable damage. Whales and sperm whales are the main animals that entangle themselves in cables and damage them. The encounter between these animals and cables can cause injury and sometimes death. Studies carried out between 1877 and 1955 reported 16 cable ruptures caused by whale entanglement, 13 of them by sperm whales. Between 1907 and 2006, 39 such events were recorded. Cable burial techniques are gradually being introduced to prevent such incidents.

The risk of fishing
Although submarine cables are located on the seabed, fishing activity can damage the cables. Fishermen using fishing techniques that involve scraping the seabed, or dragging equipment such as trawls or cages, can damage the cables, resulting in the loss of liquids and the chemical and toxic materials that make up the cables.

Areas with a high density of submarine cables have the advantage of being safer from fishing. At the expense of benthic and sedimentary zones, marine fauna is better protected in these maritime regions, thanks to limitations and bans. Studies have shown a positive effect on the fauna surrounding cable installation zones.

Pollution
Submarine cables are made of copper or optical fibers, surrounded by several protective layers of plastic, wire or synthetic materials. Cables can also be composed of dielectric fluids or hydrocarbon fluids, which act as electrical insulators. These substances can be harmful to marine life.

Fishing, aging cables and marine species that collide with or become entangled in cables can damage cables and spread toxic and harmful substances into the sea. However, the impact of submarine cables is limited compared with other sources of ocean pollution.

There is also a risk of releasing pollutants buried in sediments. When sediments are re-suspended due to the installation of cables, toxic substances such as hydrocarbons may be released.

Preliminary analyses can assess the level of sediment toxicity and select a cable route that avoids the remobilization and dispersion of sediment pollutants. And new, more modern techniques will make it possible to use less polluting materials for cable construction.

Sound waves and electromagnetic waves
The installation and maintenance of cables requires the use of machinery and equipment that can trigger sound waves or electromagnetic waves that can disturb animals that use waves to find their bearings in space or to communicate. Underwater sound waves depend on the equipment used, the characteristics of the seabed area where the cables are located, and the relief of the area.

Underwater noise and waves can modify the behavior of certain underwater species, such as migratory behavior, disrupting communication or reproduction.

Articles

 * History of the Atlantic Cable & Submarine Telegraphy – Wire Rope and the Submarine Cable Industry
 * Mother Earth Mother Board – Wired article by Neal Stephenson about submarine cables
 * Winkler, Jonathan Reed. Nexus: Strategic Communications and American Security in World War I. (Cambridge, MA: Harvard University Press, 2008) Account of how U.S. government discovered strategic significance of communications lines, including submarine cables, during World War I.
 * Animations from Alcatel showing how submarine cables are installed and repaired
 * Work begins to repair severed net
 * Flexibility in Undersea Networks – Ocean News & Technology magazine Dec. 2014
 * Work begins to repair severed net
 * Flexibility in Undersea Networks – Ocean News & Technology magazine Dec. 2014

Maps

 * Submarine Cable Map by TeleGeography
 * Map gallery of submarine cable maps by TeleGeography, showing evolution since 2000. 2008 map in the Guardian; 2014 map on CNN.
 * Map and Satellite views of US landing sites for transatlantic cables
 * Map and Satellite views of US landing sites for transpacific cables
 * Positions and Route information of Submarine Cables in the Seas Around the UK

Sjøkabel

title: Submarine communications cable