En matemáticas , una serie de potencias formales es una generalización de un polinomio , donde se permite que el número de términos sea infinito, sin requisitos de convergencia. Por lo tanto, la serie ya no puede representar una función de su variable, simplemente una secuencia formal de coeficientes, en contraste con una serie de potencias , que define una función tomando valores numéricos para la variable dentro de un radio de convergencia. En una serie de potencias formales, las potencias de la variable se utilizan solo como tenedores de posición para los coeficientes, de modo que el coeficiente dees el quinto término de la secuencia. En combinatoria , el método de generación de funciones utiliza series de potencias formales para representar secuencias numéricas y conjuntos múltiples , por ejemplo, permitiendo expresiones concisas para secuencias definidas de forma recursiva independientemente de si la recursividad se puede resolver explícitamente. De manera más general, las series de potencias formales pueden incluir series con cualquier número finito (o contable) de variables y con coeficientes en un anillo arbitrario .
En geometría algebraica y álgebra conmutativa , los anillos de series formales de potencia son anillos locales topológicamente completos especialmente tratables , lo que permite argumentos similares al cálculo dentro de un marco puramente algebraico. Son análogos en muchos aspectos a los números p-ádicos . Se pueden crear series de potencias formales a partir de polinomios de Taylor utilizando módulos formales .
Introducción
Una serie de potencias formales se puede considerar vagamente como un objeto que es como un polinomio , pero con infinitos términos. Alternativamente, para aquellos familiarizados con las series de potencias (o series de Taylor ), uno puede pensar en una serie de potencias formal como una serie de potencias en la que ignoramos cuestiones de convergencia al no asumir que la variable X denota ningún valor numérico (ni siquiera un valor desconocido ). Por ejemplo, considere la serie
Si estudiamos esto como una serie de potencias, sus propiedades incluirían, por ejemplo, que su radio de convergencia es 1. Sin embargo, como serie de potencias formal, podemos ignorar esto por completo; todo lo que es relevante es la secuencia de coeficientes [1, −3, 5, −7, 9, −11, ...]. En otras palabras, una serie de potencias formales es un objeto que simplemente registra una secuencia de coeficientes. Es perfectamente aceptable considerar una serie de potencias formales con los factoriales [1, 1, 2, 6, 24, 120, 720, 5040, ...] como coeficientes, aunque la serie de potencias correspondiente diverja para cualquier valor distinto de cero de X .
La aritmética en series formales de potencias se lleva a cabo simplemente pretendiendo que las series son polinomios. Por ejemplo, si
luego agregamos A y B término por término:
Podemos multiplicar series de potencias formales, nuevamente tratándolas como polinomios (ver en particular el producto de Cauchy ):
Tenga en cuenta que cada coeficiente en el producto AB solamente depende de un finito número de coeficientes de A y B . Por ejemplo, el término X 5 viene dado por
Por esta razón, uno puede multiplicar series de potencias formales sin preocuparse por las cuestiones habituales de convergencia absoluta , condicional y uniforme que surgen al tratar con series de potencias en el marco del análisis .
Una vez que hemos definido la multiplicación para series de potencias formales, podemos definir los inversos multiplicativos de la siguiente manera. El inverso multiplicativo de una serie de potencias formales A es una serie de potencias formales C tal que AC = 1, siempre que exista tal serie de potencias formales. Resulta que si A tiene un inverso multiplicativo, es único y lo denotamos por A −1 . Ahora podemos definir la división de series de potencias formales definiendo B / A como el producto BA −1 , siempre que exista la inversa de A. Por ejemplo, se puede usar la definición de multiplicación anterior para verificar la fórmula familiar
Una operación importante en las series de potencias formales es la extracción de coeficientes. En su forma más básica, el operador de extracción de coeficientes aplicado a una serie de potencias formales en una variable extrae el coeficiente de la la potencia de la variable, de modo que y . Otros ejemplos incluyen
De manera similar, muchas otras operaciones que se llevan a cabo en polinomios pueden extenderse a la configuración formal de series de potencias, como se explica a continuación.
El anillo de la serie formal de poder
Si se considera el conjunto de todas las series formales de potencia en X con coeficientes en un anillo conmutativo R , los elementos de este conjunto constituyen colectivamente otro anillo que se escribey llamado el anillo de series formales en la variable X sobre R .
Definición del anillo formal de la serie de potencias
Uno puede caracterizar abstractamente como la terminación del anillo polinomialequipado con una métrica particular . Esto da automáticamentela estructura de un anillo topológico (e incluso de un espacio métrico completo). Pero la construcción general de una terminación de un espacio métrico es más complicada de lo que se necesita aquí, y haría que las series de potencias formales parezcan más complicadas de lo que son. Es posible describir más explícitamente, y defina la estructura del anillo y la estructura topológica por separado, como sigue.
Estructura de anillo
Como conjunto, se puede construir como el conjunto de todas las secuencias infinitas de elementos de , indexado por los números naturales (tomado para incluir 0). Designación de una secuencia cuyo término en el índice es por , se define la suma de dos de tales secuencias por
y multiplicación por
Este tipo de producto se denomina producto de Cauchy de las dos secuencias de coeficientes y es una especie de convolución discreta . Con estas operaciones, se convierte en un anillo conmutativo con elemento cero e identidad multiplicativa .
De hecho, el producto es el mismo que se usa para definir el producto de polinomios en uno indeterminado, lo que sugiere usar una notación similar. Uno incrusta dentro enviando cualquier (constante) a la secuencia y designa la secuencia por ; luego, usando las definiciones anteriores, cada secuencia con solo un número finito de términos distintos de cero puede expresarse en términos de estos elementos especiales como
estos son precisamente los polinomios en . Dado esto, es bastante natural y conveniente designar una secuencia general por la expresión formal , aunque esta última no es una expresión formada por las operaciones de suma y multiplicación definidas anteriormente (a partir de las cuales solo se pueden construir sumas finitas). Esta convención de notación permite reformular las definiciones anteriores como
y
lo cual es bastante conveniente, pero uno debe ser consciente de la distinción entre suma formal (una mera convención) y adición real.
Estructura topológica
Habiendo estipulado convencionalmente que
( 1 )
uno quisiera interpretar el lado derecho como una suma infinita bien definida. Con ese fin, una noción de convergencia enestá definida y una topología enesta construido. Hay varias formas equivalentes de definir la topología deseada.
- Podemos dar la topología del producto , donde cada copia dese le da la topología discreta .
- Podemos dar la topología I-ádica , donde es el ideal generado por , que consta de todas las secuencias cuyo primer término es cero.
- La topología deseada también podría derivarse de la siguiente métrica . La distancia entre secuencias distintas se define como
- dónde es el número natural más pequeño tal que ; la distancia entre dos secuencias iguales es, por supuesto, cero.
De manera informal, dos secuencias y llegar a estar más y más cerca si y sólo si más y más de sus términos coinciden exactamente. Formalmente, la secuencia de sumas parciales de alguna suma infinita converge si para cada potencia fija deel coeficiente se estabiliza: hay un punto más allá del cual todas las sumas parciales adicionales tienen el mismo coeficiente. Este es claramente el caso del lado derecho de ( 1 ), independientemente de los valores, desde la inclusión del término para da el último (y de hecho único) cambio al coeficiente de . También es obvio que el límite de la secuencia de sumas parciales es igual al lado izquierdo.
Esta estructura topológica, junto con las operaciones de anillo descritas anteriormente, forman un anillo topológico. Esto se llama el anillo de la serie de poder formal sobre y se denota por . La topología tiene la propiedad útil de que una suma infinita converge si y solo si la secuencia de sus términos converge a 0, lo que simplemente significa que cualquier potencia fija de ocurre sólo en un número finito de términos.
La estructura topológica permite un uso mucho más flexible de sumas infinitas. Por ejemplo, la regla para la multiplicación se puede reformular simplemente como
ya que solo un número finito de términos de la derecha afectan a cualquier . Los productos infinitos también se definen por la estructura topológica; se puede ver que un producto infinito converge si y solo si la secuencia de sus factores converge a 1.
Topologías alternativas
La topología anterior es la mejor topología para la que
siempre converge como una suma a la serie de potencias formales designada por la misma expresión, ya menudo basta para dar un significado a sumas y productos infinitos, u otros tipos de límites que se quieran utilizar para designar series de potencias formales particulares. Sin embargo, ocasionalmente puede suceder que uno desee utilizar una topología más burda, de modo que ciertas expresiones se vuelvan convergentes que de otro modo divergirían. Esto se aplica en particular cuando el anillo de base ya viene con una topología distinta a la discreta, por ejemplo si también se trata de un anillo de series de potencias formales.
En el anillo de la serie formal de poder , la topología de la construcción anterior solo se relaciona con lo indeterminado , ya que la topología que se puso en ha sido reemplazada por la topología discreta al definir la topología de todo el anillo. Entonces
converge (y su suma se puede escribir como ); sin emabargo
se consideraría divergente, ya que cada término afecta el coeficiente de . Esta asimetría desaparece si la serie de potencia suena en se le da la topología del producto donde cada copia de se le da su topología como un anillo de series de potencias formales en lugar de la topología discreta. Con esta topología, una secuencia de elementos de converge si el coeficiente de cada potencia de converge a una serie de potencias formales en , una condición más débil que estabilizarse por completo. Por ejemplo, con esta topología, en el segundo ejemplo dado arriba, el coeficiente deconverge a , por lo que toda la suma converge a .
Esta forma de definir la topología es de hecho la estándar para construcciones repetidas de anillos de series formales de potencias, y proporciona la misma topología que se obtendría tomando series formales de potencias en todos los indeterminados a la vez. En el ejemplo anterior, eso significaría construir y aquí una secuencia converge si y solo si el coeficiente de cada monomio estabiliza. Esta topología, que también es la-topología ádica, donde es el ideal generado por y , todavía disfruta de la propiedad de que una suma converge si y solo si sus términos tienden a 0.
El mismo principio podría usarse para hacer converger otros límites divergentes. Por ejemplo en el límite
no existe, por lo que en particular no converge a
Esto es porque para el coeficiente de no se estabiliza como . Sin embargo, converge en la topología habitual de, y de hecho al coeficiente de . Por tanto, si uno diera la topología del producto de donde la topología de es la topología habitual en lugar de la discreta, entonces el límite anterior convergería a . Sin embargo, este enfoque más permisivo no es el estándar cuando se consideran series de poderes formales, ya que conduciría a consideraciones de convergencia que son tan sutiles como lo son en el análisis , mientras que la filosofía de las series de poderes formales, por el contrario, hace que las preguntas de convergencia sean tan triviales como lo son en el análisis . posiblemente pueden serlo. Con esta topología no sería el caso de que una suma converja si y solo si sus términos tienden a 0.
Universal property
The ring may be characterized by the following universal property. If is a commutative associative algebra over , if is an ideal of such that the -adic topology on is complete, and if is an element of , then there is a unique with the following properties:
- is an -algebra homomorphism
- is continuous
- .
Operaciones sobre series formales de potencias
One can perform algebraic operations on power series to generate new power series.[1][2] Besides the ring structure operations defined above, we have the following.
Power series raised to powers
For any natural number n we have
where
(This formula can only be used if m and a0 are invertible in the ring of coefficients.)
In the case of formal power series with complex coefficients, the complex powers are well defined at least for series f with constant term equal to 1. In this case, can be defined either by composition with the binomial series (1+x)α, or by composition with the exponential and the logarithmic series, or as the solution of the differential equation with constant term 1, the three definitions being equivalent. The rules of calculus and easily follow.
Multiplicative inverse
The series
is invertible in if and only if its constant coefficient is invertible in . This condition is necessary, for the following reason: if we suppose that has an inverse then the constant term of is the constant term of the identity series, i.e. it is 1. This condition is also sufficient; we may compute the coefficients of the inverse series via the explicit recursive formula
An important special case is that the geometric series formula is valid in :
If is a field, then a series is invertible if and only if the constant term is non-zero, i.e. if and only if the series is not divisible by . This means that is a discrete valuation ring with uniformizing parameter .
Division
The computation of a quotient
assuming the denominator is invertible (that is, is invertible in the ring of scalars), can be performed as a product and the inverse of , or directly equating the coefficients in :
Extracting coefficients
The coefficient extraction operator applied to a formal power series
in X is written
and extracts the coefficient of Xm, so that
Composition
Given formal power series
one may form the composition
where the coefficients cn are determined by "expanding out" the powers of f(X):
Here the sum is extended over all (k, j) with and with
A more explicit description of these coefficients is provided by Faà di Bruno's formula, at least in the case where the coefficient ring is a field of characteristic 0.
Composition is only valid when has no constant term, so that each depends on only a finite number of coefficients of and . In other words, the series for converges in the topology of .
Example
Assume that the ring has characteristic 0 and the nonzero integers are invertible in . If we denote by the formal power series
then the expression
makes perfect sense as a formal power series. However, the statement
is not a valid application of the composition operation for formal power series. Rather, it is confusing the notions of convergence in and convergence in ; indeed, the ring may not even contain any number with the appropriate properties.
Composition inverse
Whenever a formal series
has f0 = 0 and f1 being an invertible element of R, there exists a series
that is the composition inverse of , meaning that composing with gives the series representing the identity function . The coefficients of may be found recursively by using the above formula for the coefficients of a composition, equating them with those of the composition identity X (that is 1 at degree 1 and 0 at every degree greater than 1). In the case when the coefficient ring is a field of characteristic 0, the Lagrange inversion formula (discussed below) provides a powerful tool to compute the coefficients of g, as well as the coefficients of the (multiplicative) powers of g.
Formal differentiation
Given a formal power series
we define its formal derivative, denoted Df or f ′, by
The symbol D is called the formal differentiation operator. This definition simply mimics term-by-term differentiation of a polynomial.
This operation is R-linear:
for any a, b in R and any f, g in Additionally, the formal derivative has many of the properties of the usual derivative of calculus. For example, the product rule is valid:
and the chain rule works as well:
whenever the appropriate compositions of series are defined (see above under composition of series).
Thus, in these respects formal power series behave like Taylor series. Indeed, for the f defined above, we find that
where Dk denotes the kth formal derivative (that is, the result of formally differentiating k times).
Formal antidifferentiation
If is a ring with characteristic zero and the nonzero integers are invertible in , then given a formal power series
we define its formal antiderivative or formal indefinite integral by
for any constant .
This operation is R-linear:
for any a, b in R and any f, g in Additionally, the formal antiderivative has many of the properties of the usual antiderivative of calculus. For example, the formal antiderivative is the right inverse of the formal derivative:
for any .
Propiedades
Algebraic properties of the formal power series ring
is an associative algebra over which contains the ring of polynomials over ; the polynomials correspond to the sequences which end in zeros.
The Jacobson radical of is the ideal generated by and the Jacobson radical of ; this is implied by the element invertibility criterion discussed above.
The maximal ideals of all arise from those in in the following manner: an ideal of is maximal if and only if is a maximal ideal of and is generated as an ideal by and .
Several algebraic properties of are inherited by :
- if is a local ring, then so is (with the set of non units the unique maximal ideal),
- if is Noetherian, then so is (a version of the Hilbert basis theorem),
- if is an integral domain, then so is , and
- if is a field, then is a discrete valuation ring.
Topological properties of the formal power series ring
The metric space is complete.
The ring is compact if and only if R is finite. This follows from Tychonoff's theorem and the characterisation of the topology on as a product topology.
Weierstrass preparation
The ring of formal power series with coefficients in a complete local ring satisfies the Weierstrass preparation theorem.
Aplicaciones
Formal power series can be used to solve recurrences occurring in number theory and combinatorics. For an example involving finding a closed form expression for the Fibonacci numbers, see the article on Examples of generating functions.
One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting. Consider for instance the following elements of :
Then one can show that
The last one being valid in the ring
For K a field, the ring is often used as the "standard, most general" complete local ring over K in algebra.
Interpretación de series de potencias formales como funciones
In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series over certain special rings can also be interpreted as functions, but one has to be careful with the domain and codomain. Let
and suppose S is a commutative associative algebra over R, I is an ideal in S such that the I-adic topology on S is complete, and x is an element of I. Define:
This series is guaranteed to converge in S given the above assumptions on x. Furthermore, we have
and
Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved.
Since the topology on is the (X)-adic topology and is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients (so that they belong to the ideal (X)): f(0), f(X2−X) and f((1−X)−1 − 1) are all well defined for any formal power series
With this formalism, we can give an explicit formula for the multiplicative inverse of a power series f whose constant coefficient a = f(0) is invertible in R:
If the formal power series g with g(0) = 0 is given implicitly by the equation
where f is a known power series with f(0) = 0, then the coefficients of g can be explicitly computed using the Lagrange inversion formula.
Generalizaciones
Formal Laurent series
The formal Laurent series over a ring are defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree. That is, they are the series that can be written as
for some integer N, so that there are only finitely many negative n with . (This is different from the classical Laurent series of complex analysis.) For a non-zero formal Laurent series, the minimal integer such that is called the order of and is denoted (The order of the zero series is .)
Multiplication of such series can be defined. Indeed, similarly to the definition for formal power series, the coefficient of Xk of two series with respective sequences of coefficients and is
The formal Laurent series form the ring of formal Laurent series over , denoted by .[a] It is equal to the localization of with respect to the set of positive powers of . If is a field, then is in fact a field, which may alternatively be obtained as the field of fractions of the integral domain .
As with the ring of formal power series, the ring of formal Laurent series may be endowed with the structure of a topological ring by introducing the metric
One may define formal differentiation for formal Laurent series in the natural (term-by-term) way. Precisely, the formal derivative of the formal Laurent series above is
Formal residue
Assume that is a field of characteristic 0. Then the map
is a -derivation that satisfies
The latter shows that the coefficient of in is of particular interest; it is called formal residue of and denoted . The map
is -linear, and by the above observation one has an exact sequence
Some rules of calculus. As a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any
- i.
- ii.
- iii.
- iv. if
- v.
Property (i) is part of the exact sequence above. Property (ii) follows from (i) as applied to . Property (iii): any can be written in the form , with and : then implies is invertible in whence Property (iv): Since we can write with . Consequently, and (iv) follows from (i) and (iii). Property (v) is clear from the definition.
The Lagrange inversion formula
As mentioned above, any formal series with f0 = 0 and f1 ≠ 0 has a composition inverse The following relation between the coefficients of gn and f−k holds ("Lagrange inversion formula"):
In particular, for n = 1 and all k ≥ 1,
Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting it here. Noting , we can apply the rules of calculus above, crucially Rule (iv) substituting , to get:
Generalizations. One may observe that the above computation can be repeated plainly in more general settings than K((X)): a generalization of the Lagrange inversion formula is already available working in the -modules where α is a complex exponent. As a consequence, if f and g are as above, with , we can relate the complex powers of f / X and g / X: precisely, if α and β are non-zero complex numbers with negative integer sum, then
For instance, this way one finds the power series for complex powers of the Lambert function.
Power series in several variables
Formal power series in any number of indeterminates (even infinitely many) can be defined. If I is an index set and XI is the set of indeterminates Xi for i∈I, then a monomial Xα is any finite product of elements of XI (repetitions allowed); a formal power series in XI with coefficients in a ring R is determined by any mapping from the set of monomials Xα to a corresponding coefficient cα, and is denoted . The set of all such formal power series is denoted and it is given a ring structure by defining
and
Topology
The topology on is such that a sequence of its elements converges only if for each monomial Xα the corresponding coefficient stabilizes. If I is finite, then this the J-adic topology, where J is the ideal of generated by all the indeterminates in XI. This does not hold if I is infinite. For example, if then the sequence with does not converge with respect to any J-adic topology on R, but clearly for each monomial the corresponding coefficient stabilizes.
As remarked above, the topology on a repeated formal power series ring like is usually chosen in such a way that it becomes isomorphic as a topological ring to
Operations
All of the operations defined for series in one variable may be extended to the several variables case.
- A series is invertible if and only if its constant term is invertible in R.
- The composition f(g(X)) of two series f and g is defined if f is a series in a single indeterminate, and the constant term of g is zero. For a series f in several indeterminates a form of "composition" can similarly be defined, with as many separate series in the place of g as there are indeterminates.
In the case of the formal derivative, there are now separate partial derivative operators, which differentiate with respect to each of the indeterminates. They all commute with each other.
Universal property
In the several variables case, the universal property characterizing becomes the following. If S is a commutative associative algebra over R, if I is an ideal of S such that the I-adic topology on S is complete, and if x1, ..., xr are elements of I, then there is a unique map with the following properties:
- Φ is an R-algebra homomorphism
- Φ is continuous
- Φ(Xi) = xi for i = 1, ..., r.
Non-commuting variables
The several variable case can be further generalised by taking non-commuting variables Xi for i ∈ I, where I is an index set and then a monomial Xα is any word in the XI; a formal power series in XI with coefficients in a ring R is determined by any mapping from the set of monomials Xα to a corresponding coefficient cα, and is denoted . The set of all such formal power series is denoted R«XI», and it is given a ring structure by defining addition pointwise
and multiplication by
where · denotes concatenation of words. These formal power series over R form the Magnus ring over R.[3][4]
On a semiring
Given an alphabet and a semiring . The formal power series over supported on the language is denoted by . It consists of all mappings , where is the free monoid generated by the non-empty set .
The elements of can be written as formal sums
where denotes the value of at the word . The elements are called the coefficients of .
For the support of is the set
A series where every coefficient is either or is called the characteristic series of its support.
The subset of consisting of all series with a finite support is denoted by and called polynomials.
For and , the sum is defined by
The (Cauchy) product is defined by
The Hadamard product is defined by
And the products by a scalar and by
- and , respectively.
With these operations and are semirings, where is the empty word in .
These formal power series are used to model the behavior of weighted automata, in theoretical computer science, when the coefficients of the series are taken to be the weight of a path with label in the automata.[5]
Replacing the index set by an ordered abelian group
Suppose is an ordered abelian group, meaning an abelian group with a total ordering respecting the group's addition, so that if and only if for all . Let I be a well-ordered subset of , meaning I contains no infinite descending chain. Consider the set consisting of
for all such I, with in a commutative ring , where we assume that for any index set, if all of the are zero then the sum is zero. Then is the ring of formal power series on ; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same. Sometimes the notation is used to denote .[6]
Various properties of transfer to . If is a field, then so is . If is an ordered field, we can order by setting any element to have the same sign as its leading coefficient, defined as the least element of the index set I associated to a non-zero coefficient. Finally if is a divisible group and is a real closed field, then is a real closed field, and if is algebraically closed, then so is .
This theory is due to Hans Hahn, who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.
- Bell series are used to study the properties of multiplicative arithmetic functions
- Formal groups are used to define an abstract group law using formal power series
- Puiseux series are an extension of formal Laurent series, allowing fractional exponents
- Rational series
Ver también
- Ring of restricted power series
Notas
- ^ For each nonzero formal Laurent series, the order is an integer (that is, the degrees of the terms are bounded below). But the ring contains series of all orders.
Referencias
- ^ Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "0.313". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 18. ISBN 978-0-12-384933-5. LCCN 2014010276. (Several previous editions as well.)
- ^ Niven, Ivan (October 1969). "Formal Power Series". American Mathematical Monthly. 76 (8): 871–889. doi:10.1080/00029890.1969.12000359.
- ^ Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 167. ISBN 978-3-540-63003-6. Zbl 0819.11044.
- ^ Moran, Siegfried (1983). The Mathematical Theory of Knots and Braids: An Introduction. North-Holland Mathematics Studies. 82. Elsevier. p. 211. ISBN 978-0-444-86714-8. Zbl 0528.57001.
- ^ Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1, p. 12
- ^ Shamseddine, Khodr; Berz, Martin (2010). "Analysis on the Levi-Civita Field: A Brief Overview" (PDF). Contemporary Mathematics. 508: 215–237. doi:10.1090/conm/508/10002. ISBN 9780821847404.
- Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007.
- Nicolas Bourbaki: Algebra, IV, §4. Springer-Verlag 1988.
Otras lecturas
- W. Kuich. Semirings and formal power series: Their relevance to formal languages and automata theory. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, Chapter 9, pages 609–677. Springer, Berlin, 1997, ISBN 3-540-60420-0
- Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1