User:Richard Pinch/sandbox

Series-parallel order

A class of partial order. A partially ordered set has a series-parallel order if it can be constructed recursively from the one-point ordered set using the operations $+$ and $*$ defined on orders as follows. Let $P_1,{<}$ and $P_2,{<}$ be disjoint ordered sets. Each of $P_1 + P_2$ and $P_1 * P_2$ has as underlying set the disjoint union $P_1 \sqcup P_2$ and an order $\prec$ where in each case $\prec$ restricted to $P_1$ and to $P_2$ gives the original order. In $P_1 + P_2$ all pairs $p_1,p_2$ with $P_1 \in P_1$ and $p_2 \in P_2$ are incomparable (parallel); in $P_1 * P_2$, then all such pairs have $p_1 \prec p_2$ (series). We note that $+$ is commutative but $*$ is not.

Each operation is an example of the ordered sum, the operation $+$ corresponding to the trivial order on the indices and $*$ corresponding to the order $1<2$ on indices.

Series-parallel graphs are characterised by being "N-free": having no subset $\{a,b,c,d\}$ with only the comparisons $a \prec b$, $a \prec d$, $c \prec d$.

Series-parallel graphs have dimension at most $2$.

The comparability graphs of series-parallel orders are the cographs.

References

• Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics and Applications 3. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 Zbl 0919.05001
• Caspard, Nathalie; Leclerc, Bruno; Monjardet, Bernard Finite ordered sets. Concepts, results and uses Encyclopedia of Mathematics and its Applications 144 Cambridge University Press (2012) ISBN 978-1-107-01369-8 Zbl 1238.06001

Cograph

component reducible graph

An unoriented graph obtained from the single-vertex graph by the operations of disjoint union and complementation. They are characterised by having no induced subgraph $P_4$, the path of length $4$.

They are the comparability graphs of series-parallel orders.

References

• Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics and Applications 3. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 Zbl 0919.05001
• Rival, Ivan (ed.) Graphs and order. The role of graphs in the theory of ordered sets and its applications Reidel (1985) Zbl 0549.00002

Polarity

A correspondence derived from a binary relation between two sets, introduced by G. Birkhoff: a special case of a Galois correspondence. Let $R$ be a relation from $A$ to $B$, equivalently a subset of $A \times B$. Define polar maps between the power sets, $F : \mathcal{P}A \rightarrow \mathcal{P}B$ and $G : \mathcal{P}B \rightarrow \mathcal{P}A$ by $$ F(U) = \{ b \in B : aRb\ \text{for all}\ a \in U \} $$ and $$ G(V) = \{ a \in A : aRb\ \text{for all}\ b \in V \} \ . $$

Make $\mathcal{P}A$, $\mathcal{P}B$ partially ordered sets by subset inclusion. Then $F$ and $G$ are order-reversing maps, and $FG$ and $GF$ are order-preserving (monotone). Indeed, $F$ and $G$ are quasi-inverse, that is, $FGF = F$ and $GFG = G$; hence $FG$ and $GF$ are closure operators.

The closed pairs $(U,V)$ with $V = F(U)$ and $U = G(V)$ may be ordered by $(U_1,V_1) \le (U_2,V_2) \Leftrightarrow U_1 \subseteq U_2 \Leftrightarrow V_1 \supseteq V_2$. This ordered set, denoted $\mathfrak{B}(A,B,R)$, is a complete lattice with $$ \bigwedge_{i \in I} (U_i,V_i) = \left({ \bigcap_{i\in I} U_i, FG\left({ \bigcup_{i \in I} V_i }\right) }\right) $$ and $$ \bigvee_{i \in I} (U_i,V_i) = \left({ GF\left({ \bigcup_{i \in I} U_i }\right), \bigcap_{i\in I} V_i }\right) \ . $$

Every complete lattice $L$ arises in this way: indeed, $L = \mathfrak{B}(L,L,{\le})$.

References

• Birkhoff, Garrett Lattice theory American Mathematical Society (1940) Zbl 0063.00402
• Davey, B.A.; Priestley, H.A. Introduction to lattices and order (2nd ed.) Cambridge University Press (2002) ISBN 978-0-521-78451-1 Zbl 1002.06001

Frame

A generalisation of the concept of topological space occurring in the theory of logic and computation.

A frame is a complete lattice $(X,{\le})$ (a lattice with all meets and joins) satisfying the frame distributivity law, that binary meets distribute over arbitrary joins: $$ x \wedge \bigvee \{ y \in Y \} = \bigvee \{ x \wedge y : y \in Y \} \ . $$

The powerset $\mathcal{P}(A)$ of a set $A$ forms a frame.

If $(X,\mathfrak{T})$ is a topological space, with $\mathfrak{T}$ the collection of open sets, then $\mathfrak{T}$ forms a subframe of $\mathcal{P}(X)$: it should be noted that whereas the join is set-theoretic union, the meet operation is given by $$ \bigwedge \{ S \} = \mathrm{Int}\left({ \bigcap \{ S \} }\right) $$ where $\mathrm{Int}$ denotes the interior.

References

[1]	Steven Vickers Topology via Logic Cambridge Tracts in Theoretical Computer Science 5 Cambridge University Press (1989) ISBN 0-521-36062-5 Zbl 0668.54001
[2]	Jonathan S. Golan Semirings and their Applications Springer (2013) ISBN 9401593337

Alexandrov topology

on a partially ordered set $(X,{\le})$

A topology which is discrete in the broad sense, that arbitrary unions and intersections of open sets are open. Define an upper set $U \subseteq X$ to be one for which $u \in U$ and $u \le x$ implies $x \in U$. The Alexandrov topology on $X$ is that for which all upper sets are open.

The Alexandrov topology makes $X$ a T0 space and the specialisation order is just the original order ${\le}$ on $X$.

References

• Johnstone, Peter T. Stone spaces Cambridge Studies in Advanced Mathematics 3 Cambridge University Press(1986) Zbl 0586.54001

Exponentiation

The algebraic and analytic operations generalising the operation of repeated multiplication in number systems.

For positive integer $n$, the operation $x \mapsto x^n$ may be defined on any system of numbers by repeated multiplication $$ x^n = x \cdot x \cdot \cdots \cdot x\ \ \ (n\,\text{times}) $$ where $\cdot$ denotes multiplication. The number $n$ is the exponent in this operation.

The repeated operations may be carried out in any order provided that multiplication is associative, $x \cdot (y \cdot z) = (x \cdot y) \cdot z$. In this case we have $$ x^{m+n} = x^m \cdot x^n \ . $$

If the operation is also commutative then we have $$ (x \cdot y)^n = x^n \cdot y^n \ . $$

We may extend the definition to non-positive integer powers by defining $x^0 = 1$ and $$ x^{-n} = \frac{1}{x^n} $$ whenever this makes sense.

We may extend the definition to rational number exponents by taking $x^{1/n}$ to be any number $y$ such that $y^n = x$: this may denote none, one or more than one number.

Positive real numbers

Exponentiation of positive real numbers by rational exponents may be defined by taking $x^{1/n}$ to be the unique positive real solution of $y^n = x$: this always exists. We thus have $x^q$ well-defined for $x>0$ and any rational exponent $q$. Exponentiation preserves order: if $x > y$ then $x^q > y^q$ if $q > 0$ and $x^q < y^q$ if $q < 0$.

We can now define exponentiation with real exponent $r$ by defining $x^r$ to be the limit of $x^{q_n}$ where $q_n$ is a sequence of rational numbers converging to $r$. There is always such a sequence, and the limit exists and does not depend on the chosen sequence.

For positive real numbers there are mutually inverse exponential and logarithm functions which allow the alternative definition $$ x^y = \exp(y \log x) \ . $$ Here the exponential function may be regarded as $\exp(x) = e^x$ where $e$ is the base of natural logarithms.

Complex numbers

Exponentiation of complex numbers with non-integer exponents may be defined using the complex exponential function and logarithm. The exponential function is analytic and defined on the whole complex plane: the logarithmic function requires a choice of branch, corresponding to a choice of the range of values for the argument, to make it single-valued. Given such a choice, exponentiation may be defined as $z^w = \exp(w \log z)$.

General algebraic systems

For a general (not necessarily associative) binary operation, it is necessary to define the order of operations. The left and right principal powers are defined inductively by $$ x^{n+1} = x \star (x^n) $$ and $$ x^{n+1} = (x^n) \star x $$ respectively. A binary operation is power associative if the powers of a single element form an associative subsystem, so that exponentiation is well-defined.

Logarithm

The operation inverse to exponentiation.

Over the fields of real or complex numbers, one speaks of the logarithm of a number. The logarithmic function is the complex analytic function inverse to the exponential function.

In a finite Abelian group, the discrete logarithm is the inverse to exponentiation, with applications in cryptography.

The Zech logarithm in a finite field is related to the discrete logarithm.

I-semigroup

A topological semigroup defined on a totally ordered set. Let $I$ be a totally ordered set with minimum element $0$ and maximum element $1$, and equipped with the order topology; then $0$ acts as a zero (absorbing) element for the semigroup operation and $1$ acts as an identity (neutral) element. Although not required by the definition, it is the case that an I-semigroup is commutative.

Examples. The real interval $[0,1]$ under multiplication. The nil interval $[\frac12,1]$ with operation $x \circ y = \max(xy,\frac12)$. The min interval $[0,1]$ with operation $x \cdot y = \min(x,y)$.

References

• Hofmann, K.H.; Lawson, J.D. "Linearly ordered semigroups: Historical origins and A. H. Clifford’s influence" in Hofmann, Karl H. (ed.) et al., Semigroup theory and its applications. Proceedings of the 1994 conference commemorating the work of Alfred H. Clifford London Math. Soc. Lecture Note Series 231 Cambridge University Press (1996) pp.15-39 Zbl 0901.06012

Composition algebra

An algebra $A$ (not necessarily associative) over a field $K$ with a quadratic form $q$ taking values in $K$ which is multiplicative, $q(x\cdot y) = q(x) q(y)$. The composition algebras over the field $\mathbf{R}$ of real numbers are the real numbers, the field of complex numbers $\mathbf{C}$, the skew-field of quaternions, the non-associative algebra of octonions.

References

• Springer, Tonny A.; Veldkamp, Ferdinand D. Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics. Springer (2000) ISBN 3-540-66337-1 Zbl 1087.17001

Cayley–Dickson process

A construction of an algebra $A_1$ from an algebra $A$ with involution over a field $K$ which generalises the construction of the complex numbers, quaternions and octonion. Fix a parameter $d \in A$. As a set $A_1 = A \times A$ with addition defined by $(a_1,a_2) + (b_1,b_2) = (a_1+b_1, a_2+b_2)$ and multiplication by $$ (a_1,a_2) \cdot (b_1,b_2) = (a_1b_1 - d b_2 a_2^* , a_1^*b_2 + b_1a_2) \ . $$ The algebra $A_1$ has an involution $(x_1,x_2) \mapsto (x_1^*,-x_2)$.

BCH code

A cyclic code over a finite field. Fix length $n$ and ground field $\mathbf{F}_q$ and a design distance parameter $\delta$. Let $\beta$ be a primitive $n$-th root of unity in a suitable extension of $\mathbf{F}_q$. The generator of the cyclic code is the least common multiple $g$ of the minimal polynomials (over $\mathbf{F}_q$) of the elements $\beta^1, \beta^2, \ldots, \beta^{\delta-1}$.

The minimum distance of the BCH code generated by $g$ is at least $\delta$: this is the BCH bound.

As an example, let $q=2$ and $\beta$ be a primitive $7$-th root of unity in $\mathrm{F}_{8}$: we may take $\beta$ to satisfy the polynomial $x^3 + x + 1$. Choose $\delta = 3$. The minimal polynomial for $\beta^2$ is the same as that of $\beta$, so that the cyclic code is generated by the word $1101000$. This is in fact the Hamming [7,4] code.

References

• Goldie, Charles M.; Pinch, Richard G.E. Communication theory, London Mathematical Society Student Texts. 20 Cambridge University Press (1991) iSBN 0-521-40456-8 Zbl 0746.94001

Hamming distance

A function on words of fixed length over an alphabet describing the number of changes to the symbols of one word required to reduce it to another. Let $A$ be an alphabet of symbols and $C$ a subset of $A^n$, the set of words of length $n$ over $A$. Let $u=(u_1,\ldots,u_n)$ and $v = (v_1,\ldots,v_n)$ be words in $C$. The Hamming distance $d(u,v)$ is defined as the number of places in which $u$ and $v$ differ: that is, $\sharp\{ i : u_i \neq v_i,\,i=1,\dots,n \}$.

The Hamming distance satisfies $$ d(u,v) \ge 0 \ \text{and}\ d(u,v) = 0\ \text{if and only if}\ u=v\ ; $$ $$ d(u,v) = d(v,u)\ ; $$ $$ d(u,v) \le d(u,w) + d(w,v) \ . $$ Hamming distance is thus a metric on $C$.

In the theory of error-correcting codes it is assumed that words from $C$ are transmitted down a noisy channel and that some symbols are changed during the transmission: see for example the symmetric channel. Maximum likelihood decoding of a received word $r$ consists of finding the word in $C$ nearest to $r$ in the Hamming metric. If the minimum Hamming distance between words of $C$ is $\delta$, then this process is capable of detecting up to $\delta-1$ errors in $r$, and correcting up to $\left\lfloor \frac{\delta-1}{2} \right\rfloor$ errors.

References

• Goldie, Charles M.; Pinch, Richard G.E. Communication theory, London Mathematical Society Student Texts. 20 Cambridge University Press (1991) iSBN 0-521-40456-8 Zbl 0746.94001

Leibniz algebra

An algebra over a field $K$ generalising the properties of a Lie algebra. A Leibniz algebra $L$ is a $K$-algebra with multiplication denoted by $[\cdot,\cdot]$ satisfying $$ [ x, [y,z]] = [[x,y],z] - [[x,z],y] \ . $$ Every Lie algebra is a Leibniz algebra, and a Leibniz algebra is a Lie aglebra if in addition $[x,x] = 0$.

The free Leibniz algebra on a generating set $X$ may be defined as the quotient of the free non-associative algebra over $K$ (cf. Free algebra over a ring) by the ideal generated by all elements of the form $[ x, [y,z]] - [[x,y],z] + [[x,z],y] $. The standard Leibniz algebra on $X$ is obtained from the vector space $V = KX$ and forming the tensor module $$ T(X) = V \oplus V^{{}\otimes 2} \oplus \cdots \oplus V^{{}\otimes n} \oplus \cdots $$ with the multiplication $$ [x,v] = x \otimes v $$ when $v \in V$ and $$ [x, y\otimes v] = [x,y] \otimes v - [x \otimes v, y] \ . $$ The standard algebra is then a presentation of the free algebra on $X$.

References

• Mikhalev, Alexander A.; Shpilrain, Vladimir; Yu, Jie-Tai, Combinatorial methods. Free groups, polynomials, and free algebras, CMS Books in Mathematics 19 Springer (2004) ISBN 0-387-40562-3 Zbl 1039.16024

Tridiagonal matrix

A matrix with non-zero entries only on the main diagonal and the diagonals immediately above and below, for example $$ \begin{pmatrix} a_1 & b_1 & 0 & 0 & \ldots & 0 & 0 \\ c_1 & a_2 & b_2 & 0 & \ldots & 0 & 0 \\ 0 & c_2 & a_3 & b_3 & \ldots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & c_{n-1} & a_n \end{pmatrix} \ . $$

The determinant of a tridiagonal matrix may be computed as a continuant.

References

• Thomas Muir. A treatise on the theory of determinants. (Dover Publications, 1960 [1933])

Free differential calculus

Let $F$ be a free group on a set of generators $X = \{x_i : i \in I \}$ and $R[F]$ the group ring of $F$ over a commutative unital ring $R$. The Fox derivative $\partial_i$ are maps from $F$ to $R[F]$ defined by $$ \partial_i(x_j) = \delta_{ij} \ , $$ $$ \partial_i(1) = 0 \ , $$ $$ \partial_i(uv) = u \partial_i(v) + \partial_i(u) v \ . $$ It follows that $$ \partial_i(x_i^{-1}) = - x_i^{-1} \ . $$

The maps extend to derivations on $R[F]$.

References

• D. L. Johnson, Presentations of Groups, London Mathematical Society Student Texts 15 Cambridge University Press (1997) ISBN 0-521-58542-2

Martin's axiom

An axiom of set theory. Let $(P,{<})$ be a partially ordered set satisfying the countable chain condition and $D$ a family of $\mathfrak{k}$ dense subsets of $P$ for $\mathfrak{k}$ a cardinal less than $2^{\aleph_0}$. Then $\text{MA}_{\mathfrak{k}}$ asserts that there is a $D$-generic filter on $P$. Martin's axiom $\text{MA}$ is the conjunction of $\text{MA}_{\mathfrak{k}}$ for all $\mathfrak{k} < 2^{\aleph_0}$.

The case $\text{MA}_{\aleph_0}$ holds in ZFC. MA is a consequence of the Continuum hypothesis ($\text{CH}$) but $\text{MA} \wedge \text{CH}$ is consistent with ZFC if ZFC is consistent.

References

• Thomas Jech, Set Theory, Perspectives in Mathematical Logic, Third Millennium Edition, revised and expanded. Springer (2007) ISBN 3-540-44761-X

Hurwitz zeta function

generalised zeta function

An Dirichlet series related to the Riemann zeta function which may be used to exhibit properties of various Dirichlet L-functions.

The Hurwitz zeta function $\zeta(\alpha,s)$ is defined for real $\alpha$, $0 < \alpha \le 1$ as $$ \zeta(\alpha,s) = \sum_{n=0}^\infty (n+\alpha)^{-s} \ . $$ The series is convergent, and defines an analytic function, for $\Re s > 1$. The function possesses an analytic continuation to the whole $s$-plane except for a simple pole of residue 1 at $s=1$.

References

• Tom M. Apostol, "Introduction to Analytic Number Theory", Undergraduate Texts in Mathematics, Springer (1976) ISBN 0-387-90163-9 Zbl 0335.10001

Isomorphism theorems

Three theorems relating to homomorphisms of general algebraic systems.

First Isomorphism Theorem

Let $f : A \rightarrow B$ be a homomorphism of $\Omega$-algebras and $Q$ the kernel of $f$, as an equivalence relation on $A$. Then $Q$ is a congruence on $A$, and $f$ can be factorised as $f = \epsilon f' \mu$ where $\epsilon : A \rightarrow A/Q$ is the quotient map, $f' : A/Q \rightarrow f(A)$ and $\mu : f(A) \rightarrow B$ is the inclusion map. The theorem asserts that $f'$ is well-defined and an isomorphism.

Second Isomorphism Theorem

Let $Q$ be a congruence on the $\Omega$-algebra $A$ and let $A_1$ be a subalgebra of $A$. The saturation $A_1^Q$ is a subalgebra of $A$, the restriction $Q_1 = Q \ \cap A_1 \times A_1$ is a congruence on $A_1$ and there is an isomorphism $$ A_1 / Q_1 \cong A_1^Q / Q \ . $$

Third Isomorphism Theorem

Let $A$ be an $\Omega$-algebra and $Q \subset R$ congruences on $A$. There is a unique homomorphism $\theta$ from $A/Q \rightarrow A/R$ compatible with the quotient maps from $A$ to $A/R$ and $A/Q$. If $R/Q$ denotes the kernel of $\theta$ on $A/Q$ then there is an isomorphism $$ (A/Q)/(R/Q) \cong A/R \ . $$

Application to groups

In the case of groups, a congruence $Q$ on $G$ is determined by the congruence class $N = [1_G]_Q$ of the identity $1_G$, which is a normal subgroup, and the other $Q$-classes are the cosets of $N$. It is conventional to write $G/N$ for $G/Q$. The saturation of a subgroup $H$ is the complex $H^Q = HN$.

First Isomorphism Theorem for groups

Let $f : A \rightarrow B$ be a homomorphism of groups and $N = f^{-1}(1_B)$ the kernel of $f$. Then $N$ is a normal subgroup of $A$, and $f$ can be factorised as $f = \epsilon f' \mu$ where $\epsilon : A \rightarrow A/N$ is the quotient map, $f' : A/N \rightarrow f(A)$ and $\mu : f(A) \rightarrow B$ is the inclusion map. The theorem asserts that $f'$ is well-defined and an isomorphism.

Second Isomorphism Theorem for groups

Let $N$ be a normal subgroup $A$ and let $A_1$ be a subgroup of $A$. The complex $NA_1$ is a subgroup of $A$, the intersection $N_1 = N \cap A_1$ is a normal subgroup o $A_1$ and there is an isomorphism $$ A_1 / N_1 \cong A_1N/N \ . $$

Third Isomorphism Theorem for groups

Let $A$ be a group and $N \subset M$ normal subgroups of $A$. There is a unique homomorphism $\theta$ from $A/N \rightarrow A/M$ compatible with the quotient maps from $A$ to $A/N$ and $A/M$. The set $M/N$ is the kernel of $\theta$ and hence a normal subgroup of $A/N$ and there is an isomorphism $$ (A/N)/(M/N) \cong A/M \ . $$

References

• Paul M. Cohn, Universal algebra, Kluwer (1981) ISBN 90-277-1213-1

Square-free

quadratfrei

Containing only a trivial square factor.

A natural number $n$ is square-free if the only natural number $d$ such that $d^2$ divides $n$ is $d=1$. The prime power factorisation of such a number $n$ has all exponents equal to 1. Similarly a polynomial $f$ is square-free if the only factors $g$ such that $g^2$ divides $f$ are constants. For polynomials, this is equivalent to having no repeated roots in any field.

An element $x$ of a monoid $M$ is square-free if the only $y \in M$ such that $y^2$ divides $x$ are units.

A word $x$ over an alphabet $A$, that is, an element of the free monoid $A^*$, is square-free if $x=uwwv$ implies that $w$ is the empty string.

Square-free number

An integer $n$ is square-free if the only natural number $d$ such that $d^2$ divides $n$ is $d=1$. The prime factorisation of such a number $n$ has all exponents equal to 1. Any integer is uniquely expressible in the form $n = k^2 m$ where $m$ is the square-free kernel of $n$. If $Q(x)$ counts the square-free natural numbers $\le x$, then $$ Q(x) = \frac{6}{\pi^2} x + o\left({ x^{1/2} }\right) \ . $$

References

• E. Landau, "Über den Zusammenhang einiger neuerer Sätze der analytischen Zahlentheorie", Wien. Ber. 115 (1906) 589-632. Zbl 37.0236.01
• József Sándor; Dragoslav S. Mitrinović; Borislav Crstici, edd. "Handbook of number theory I". Springer-Verlag (2006). Sect.VI.18. ISBN 1-4020-4215-9. Zbl 1151.11300

Square-free polynomial

A polynomial $f$ over a field is square-free if the only factors $g$ such that $g^2$ divides $f$ are constants. For polynomials, this is equivalent to having no repeated roots. Over fields of characteristic zero, a polynomial is square-free if and only if it is coprime to its formal derivative. Over fields of characteristic $p$, this holds for separable polynomials, those $f$ such that $f' \not\equiv 0$, that is, those polynomials in $X$ that are not polynomials in $X^p$.

Over a finite field $\mathbb{F}_q$, the number of square-free monic polynomials of degree $d$ is $(1-q^{-1})q^d$.

References

• Gary L. Mullen, Daniel Panario (edd), "Handbook of Finite Fields", CRC Press (2013) ISBN 1439873828

Square-free word

A word $x$ over an alphabet $A$, that is, an element of the free monoid $A^*$, is square-free if $x=uwwv$ implies that $w$ is the empty string. A square-free word is thus one that avoids the pattern $XX$.

Examples

Over a two-letter alphabet $\{a,b\}$ the only square-free words are the empty word and $a$, $b$, $ab$, $ba$m $aba$ and $bab$. However, there exist infinite square-free words in any alphabet with three or more symbols, as proved by Axel Thue.

One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet $\{0,\pm1\}$ obtained by taking the first difference of the Thue–Morse sequence.

An example found by John Leech is defined recursively over the alphabet $\{a,b,c\}$. Let \(w_1\) be any word starting with the letter $a$. Define the words \( \{w_i \mid i \in \mathbb{N} \}\) recursively as follows: the word \(w_{i+1}\) is obtained from \(w_i\) by replacing each $a$ in \(w_i\) with $abcbacbcabcba$, each $b$ with $bcacbacabcacb$, and each $c$ with $cabacbabcabac$. The sequence $(w_i)$ converges to the infinite square-free word $$ abcbacbcabcbabcacbacabcacbcabacbabcabacbcacbacabcacb \ldots $$

Related concepts

A cube-free word is one with no occurrence of $www$ for a factor $w$. The Thue–Morse sequence is an example of a cube-free word over a binary alphabet. This sequence is not square-free but is "almost" so: the critical exponent is 2. The Thue–Morse sequence has no overlap or overlapping square, instances of $0X0X0$ or $1X1X1$: it is essentially the only infinite binary word with this property.

An abelian $p$-th power is a subsequence of the form \(w_1 \cdots w_p\) where each \(w_i\) is a permutation of \(w_1\). There is no abelian-square-free infinite word over an alphabet of size three: indeed, every word of length eight over such an alphabet contains an abelian square. There is an infinite abelian-square-free word over an alphabet of size five.

References

• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V.; Combinatorics on words. Christoffel words and repetitions in words, ser. CRM Monograph Series 27 (2009), American Mathematical Society, Zbl 1161.68043URL: http://www.ams.org/bookpages/crmm-27] ISBN: 978-0-8218-4480-9
• Lothaire, M.; Combinatorics on words, (1997), Cambridge University Press ISBN: 0-521-59924-5.
• Lothaire, M.; Algebraic combinatorics on words, ser. Encyclopedia of Mathematics and Its Applications 90 (2011), Cambridge University Press, Zbl 1221.68183 ISBN: 978-0-521-18071-9
• Pytheas Fogg, N.; Substitutions in dynamics, arithmetics and combinatorics, ser. Lecture Notes in Mathematics 1794 (2002), Springer-Verlag, Zbl 1014.11015 ISBN: 3-540-44141-7

Root of unity

An element $\zeta$ of a ring $R$ with unity $1$ such that $\zeta^m = 1$ for some $m \ge 1$. The least such $m$ is the order of $\zeta$.

A primitive root of unity of order $m$ is an element $\zeta$ such that $\zeta^m = 1$ and $\zeta^r \neq 1$ for any positive integer $r < m$. The element $\zeta$ generates the cyclic group $\mu_m$ of roots of unity of order $m$. For any $k$ that is relatively prime to $m$, the element $\zeta^k$ is also a primitive root. The number of all primitive roots of order $m$ is equal to the value of the Euler function $\phi(m)$ if $\mathrm{hcf}(m,\mathrm{char}(K)) = 1$. The primitive roots of unity are the roots of the cyclotomic polynomial of order $m$.

If the field $K$ contains a primitive root of unity of order $m$, then $m$ is relatively prime to the characteristic of $K$. An algebraically closed field contains a primitive root of any order that is relatively prime with its characteristic. In the field of complex numbers, there are roots of unity of every order: those of order $m$ take the form $$ \cos \frac{2\pi k}{m} + i \sin \frac{2\pi k}{m} $$ where $0 < k < m$ and $k$ is relatively prime to $m$.

References

[1]	S. Lang, "Algebra" , Addison-Wesley (1984)

Law of quadratic reciprocity

Gauss reciprocity law

A relation connecting the values of the Legendre symbols (cf. Legendre symbol) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. Quadratic reciprocity law). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation

$$\left(\frac pq\right)\left(\frac qp\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$

there are two more additions to this law, viz.:

$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}\quad\text{and}\quad\left(\frac2p\right)=(-1)^{(p^2-1)/8}.$$

The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to give a complete proof of the law [1]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime.

Attempts to establish the reciprocity law for cubic and biquadratic residues led Gauss to introduce the ring of Gaussian integers.

References

[1]	C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)
[2]	I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)
[3]	H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)

Comments

Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of algebraic number theory and class field theory. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law.

Quadratic reciprocity law

The relation

$$\left(\frac pq\right)\left(\frac pq\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$

connecting the Legendre symbols (cf. Legendre symbol)

$$\left(\frac pq\right)\quad\text{and}\quad\left(\frac qp\right)$$

for different odd prime numbers $p$ and $q$. There are two additions to this quadratic reciprocity law, namely:

$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}$$

and

$$\left(\frac 2p\right)=(-1)^{(p^2-1)/8}.$$

C.F. Gauss gave the first complete proof of the quadratic reciprocity law, which for this reason is also called the Gauss reciprocity law.

It immediately follows from this law that for a given square-free number $d$, the primes $p$ for which $d$ is a quadratic residue modulo $p$ ly in certain arithmetic progressions with common difference $2|d|$ or $4|d|$. The number of these progressions is $\phi(2|d|)/2$ or $\phi(4|d|)/2$, where $\phi(n)$ is the Euler function. The quadratic reciprocity law makes it possible to establish factorization laws in quadratic extensions $\mathbf Q(\sqrt d)$ of the field of rational numbers, since the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a prime number that does not divide $d$ depends on whether or not $x^2-d$ is reducible modulo $p$.

References

[1]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)

Comments

See also Quadratic residue; Dirichlet character.

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII

Euler function

Euler function

The arithmetic function $\phi$ whose value at $n$ is equal to the number of positive integers not exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The Euler function is a multiplicative arithmetic function, that is $\phi(1)=1$ and $\phi(mn)=\phi(m)\phi(n)$ for $(m,n)=1$. The function $\phi(n)$ satisfies the relations

$$\sum_{d|n}\phi(d)=n,$$

$$c\frac{n}{\ln\ln n}\leq\phi(n)\leq n,$$

$$\sum_{n\leq x}\phi(n)=\frac{3}{\pi^2}x^2+O(x\ln x).$$

It was introduced by L. Euler (1763).

The function $\phi(n)$ can be evaluated by $\phi(n)=n\prod_{p|n}(1-p^{-1})$, where the product is taken over all primes dividing $n$, cf. [a1].

For a derivation of the asymptotic formula in the article above, as well as of the formula

$$\lim_{n\to\infty}\inf\phi(n)\frac{\ln\ln n}{n}=e^{-\gamma},$$

where $\gamma$ is the Euler constant, see also [a1], Chapts. 18.4 and 18.5.

The Carmichael conjecture on the Euler totient function states that if $\phi(x) = m$ for some $m$, then $\phi(y) = m$ for some $y \neq x$; i.e. no value of the Euler function is assumed once. This has been verified for $x < 10^{1000000}$, [c1].

D. H. Lehmer asked whether whether there is any composite number $n$ such that $\phi(n)$ divides $n-1$. This is true of every prime number, and Lehmer conjectured in 1932 that there are no composite numbers with this property: he showed that if any such $n$ exists, it must be odd, square-free, and divisible by at least seven primes [c2]. For some results on this still (1996) largely open problem, see [c3] and the references therein.

References

[1]	K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) MR0249348 Zbl 0169.37502
[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 MR0568909 Zbl 0423.10001
[c1]	A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" Math. Comp. , 63 (1994) pp. 415–419
[c2]	D.H. Lehmer, "On Euler's totient function", Bulletin of the American Mathematical Society 38 (1932) 745–751 DOI 10.1090/s0002-9904-1932-05521-5 Zbl 0005.34302
[c3]	V. Siva Rama Prasad, M. Rangamma, "On composite $n$ for which $\phi(n)|n-1$" Nieuw Archief voor Wiskunde (4) , 5 (1987) pp. 77–83
[c4]	M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" Rocky Mount. J. Math. , 15 (1985) pp. 609–620
[c5]	R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" Nieuw Archief Wiskunde (4) , 8 (1990) pp. 169–188
[c6]	R. Sivamarakrishnan, "The many facets of Euler's totient I" Nieuw Archief Wiskunde (4) , 4 (1986) pp. 175–190
[c7]	L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155

Totient function

Euler totient function, Euler totient

Another frequently used named for the Euler function $\phi(n)$, which counts a reduced system of residues modulo $n$: the natural numbers $k \in \{1,\ldots,n\}$ that are relatively prime to $n$.

A natural generalization of the Euler totient function is the Jordan totient function $J_k(n)$, which counts the number of $k$-tuples $(a_1,\ldots,a_k)$, $a_i \in \{1,\ldots,n\}$, such that $\mathrm{hcf}\{n,a_1,\ldots,a_k\} = 1$. Clearly, $J_1 = \phi$. The $J_k$ are multiplicative arithmetic functions.

One has $$ J_k(n) = n^k \prod_{p|n} \left({ 1 - p^{-k} }\right) $$ where $p$ runs over the prime numbers dividing $n$, and $$ J_k(n) = \sum_{d | n} \mu(n/d) d^k $$ where $\mu$ is the Möbius function and $d$ runs over all divisors of $n$. For $k=1$ these formulae reduce to the well-known formulae for the Euler function.

References

[a1]	A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" Math. Comp. , 63 (1994) pp. 415–419
[a2]	D.H. Lehmer, "On Euler's totient function" Bull. Amer. Math. Soc. , 38 (1932) pp. 745–751
[a3]	V. Siva Rama Prasad, M. Rangamma, "On composite $n$ for which $\phi(n)|n-1$" Nieuw Archief voor Wiskunde (4) , 5 (1987) pp. 77–83
[a4]	M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" Rocky Mount. J. Math. , 15 (1985) pp. 609–620
[a5]	R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" Nieuw Archief Wiskunde (4) , 8 (1990) pp. 169–188
[a6]	R. Sivamarakrishnan, "The many facets of Euler's totient I" Nieuw Archief Wiskunde (4) , 4 (1986) pp. 175–190
[a7]	L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155

Jordan totient function

An arithmetic function $J_k(n)$ of a natural number $n$, named after Camille Jordan, counting the $k$-tuples of positive integers all less than or equal to $n$ that form a coprime $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's totient function, which is $J_1$.

Jordan's totient function is multiplicative and may be evaluated as $$ J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) \ . $$

By Möbius inversion we have $\sum_{d | n } J_k(d) = n^k $. The average order of $J_k(n)$ is $c n^k$ for some $c$.

The analogue of Lehmer's problem for the Jordan totient function (and $k>1$) is easy, [c4]: For $k>1$, $J_k(n) | n^k-1$ if and only if $n$ is a prime number. Moreover, if $n$ is a prime number, then $J_k(n) = n^k-1$. For much more information on the Euler totient function, the Jordan totient function and various other generalizations, see [c5], [c6].

References

• Dickson, L.E. History of the Theory of Numbers I, Chelsea (1971) p. 147, ISBN 0-8284-0086-5
• Ram Murty, M. Problems in Analytic Number Theory, Graduate Texts in Mathematics 206 Springer-Verlag (2001) p. 11 ISBN 0387951431 Zbl 0971.11001
• Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.32–36. ISBN 1-4020-2546-7. Zbl 1079.11001

Multiplicative sequence

Also m-sequence, a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

Definition

Let $K_n$ be polynomials over a ring $A$ in indeterminates $p_1,\ldots$ weighted so that $p_i$ has weight $i$ (with $p_0=1$) and all the terms in $K_n$ have weight $n$ (so that $K_n$ is a polynomial in $p_1,\ldots,p_n$). The sequence $K_n$ is multiplicative if an identity

$$\sum_i p_i z^i = \sum p'_i z^i \cdot \sum_i p''_i z^i $$

implies

$$\sum_i K_i(p_1,\ldots,p_i) z^i = \sum_j K_j(p'_1,\ldots,p'_j) z^j \cdot \sum_k K_k(p''_1,\ldots,p''_k) z^k . $$ The power series

$$\sum K_n(1,0,\ldots,0) z^n $$

is the characteristic power series of the $K_n$. A multiplicative sequence is determined by its characteristic power series $Q(z)$, and every power series with constant term 1 gives rise to a multiplicative sequence.

To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of z^j in the product

$$ \prod_{i=1}^m Q(\beta_i z) $$

for any $m>j$. This is symmetric in the $\beta_i$ and homogeneous of weight j: so can be expressed as a polynomial $K_j(p_1,\ldots,p_j)$ in the elementary symmetric functions $p$ of the $\beta$. Then $K_j$ defines a multiplicative sequence.

Examples

As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$.

Consider the power series

$$ Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k $$ where $B_k$ is the $k$-th Bernoulli number. The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$.

The multiplicative sequence with characteristic power series

$$ Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} $$

is denoted $A_j(p_1,\ldots,p_j)$.

The multiplicative sequence with characteristic power series

$$Q(z) = \frac{z}{1-\exp(-z)} = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} $$ is denoted $T_j(p_1,\ldots,p_j)$: the Todd polynomials.

Genus

The genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

For example, the Todd genus is associated to the Todd polynomials $T_j$ with characteristic power series $$\frac{z}{1-\exp(-z)}$$ and the L-genus is associated to the polynomials $L_j$ with charac\teristic polynomial $$\frac{\sqrt z}{\tanh \sqrt z} . $$

References

• Hirzebruch, Friedrich. Topological methods in algebraic geometry, Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] (Berlin: Springer-Verlag, 1995). ISBN 3-540-58663-6. Zbl 0843.14009.

Nagao's theorem

A result, named after Hirosi Nagao, about the structure of the group of 2-by-2 invertible matrices over the ring of polynomials over a field. It has been extended by Jean-Pierre Serre to give a description of the structure of the corresponding matrix group over the coordinate ring of a projective algebraic curve.

Nagao's theorem

For a general ring $R$ we let $GL_2(R)$ denote the group of invertible 2-by-2 matrices with entries in $R$, and let $R^*$ denote the group of units of $R$, and let

$$ B(R) = \left\lbrace{ \left({\begin{array}{*{20}c} a & b \\ 0 & d \end{array}}\right) : a,d \in R^*, ~ b \in R }\right\rbrace \ . $$

Then $B(R)$ is a subgroup of $GL_2(R)$.

Nagao's theorem states that in the case that $R$ is the ring $K[t]$ of polynomials in one variable over a field $K$, the group $GL_2(R)$ is the amalgamated product of $GL_2(K)$ and $B(K[t])$ over their intersection $B(K)$.

Serre's extension

In this setting, $C$ is a smooth projective curve over a field $K$. For a closed point $P$ of $C$ let $R$ be the corresponding coordinate ring of $C$ with $P$ removed. There exists a graph of groups $(G,T)$ where $T$ is a tree with at most one non-terminal vertex, such that $GL_2(R)$ is isomorphic to the fundamental group $\pi_1(G,T)$.

References

• Mason, A.. "Serre's generalization of Nagao's theorem: an elementary approach". Transactions of the American Mathematical Society 353 (2001) 749–767. DOI 10.1090/S0002-9947-00-02707-0 Zbl 0964.20027.
• Nagao, Hirosi. "On $GL(2, K[x])$". J. Inst. Polytechn., Osaka City Univ., Ser. A 10 (1959) 117–121. MR0114866. Zbl 0092.02504.
• Serre, Jean-Pierre. Trees. (Springer, 2003) ISBN 3-540-44237-5.

Brauer–Wall group

A group classifying graded central simple algebras over a field. It was first defined by Wall (1964) as a generalisation of the Brauer group.

The Brauer group $\mathrm{B}(F)$ of a field $F$ is defined on the isomorphism classes of central simple algebras over F. The analogous construction for $\mathbf{Z}/2$-graded algebras defines the Brauer–Wall group $\mathrm{BW}(F)$.[Lam (2005) pp.98–99]

Properties

• The Brauer group $\mathrm{B}(F)$ injects into $\mathrm{BW}(F)$ by mapping a CSA $A$ to the graded algebra which is $A$ in grade zero.

There is an exact sequence $$ 0 \rightarrow \mathrm{B}(F) \rightarrow \mathrm{BW}(F) \rightarrow Q(F) \rightarrow 0 $$ where $Q(F)$ is the group of graded quadratic extensions of $F$, defined as $\mathbf{Z}/2 \times F^*/(F^*)^2$ with multiplication $(e,x)(f,y) = (e+f,(-1)^{ef}xy)$. The map from W to BW is the Clifford invariant defined by mapping an algebra to the pair consisting of its grade and determinant.

There is a map from the additive group of the Witt–Grothendieck ring to the Brauer–Wall group obtained by sending a quadratic space to its Clifford algebra. The map factors through the Witt group[Lam (2005) p.113] which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.[Lam (2005) p.115]

Examples

• $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$. This is an algebraic aspect of Bott periodicity.

References

• Lam, Tsit-Yuen, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, (American Mathematical Society, 2005) ISBN 0-8218-1095-2 MR2104929, Zbl 1068.11023
• Wall, C. T. C., "Graded Brauer groups", Journal für die reine und angewandte Mathematik 213 (1964) 187–199, ISSN 0075-4102, Zbl 0125.01904, MR0167498

Factor system

A function on a group giving the data required to construct an algebra. A factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.

Let $G$ be a group and $L$ a field on which $G$ acts as automorphisms. A cocycle or factor system is a map $c : G \times G \rightarrow L^*$ satisfying $$ c(h,k)^g c(hk,g) = c(h,kg) c(k,g) \ . $$

Cocycles $c,c'$ are equivalent if there exists some system of elements $a : G \rightarrow L^*$ with $$ c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) \ . $$

Cocycles of the form $$ c(g,h) = a_g^h a_h a_{gh}^{-1} $$ are called split. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group $H^2(G,L^*)$.

Crossed product algebras

Let us take the case that $G$ is the Galois group of a field extension $L/K$. A factor system $c$ in $H^2(G,L^*)$ gives rise to a crossed product algebra $A$, which is a $K$-algebra containing $L$ as a subfield, generated by the elements $\lambda \in L$ and $u_g$ with multiplication $$ \lambda u_g = u_g \lambda^g \ , $$ $$ u_g u_h = u_{gh} c(g,h) \ . $$ Equivalent factor systems correspond to a change of basis in $A$ over $K$. We may write $$ A = (L,G,c) \ .$$

Every central simple algebra over$K$ that splits over $L$ arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in$H^2$. We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over $K$, with $H^2$.[Saltman (1999) p.44]

Cyclic algebra

Let us further restrict to the case that $L/K$ is cyclic with Galois group $G$ of order $n$ generated by $t$. Let $A$ be a crossed product $(L,G,c)$ with factor set $c$. Let $u=u_t$ be the generator in $A$ corresponding to $t$. We can define the other generators $$ u_{t^i} = u^i $$ and then we have $u^n = a$ in $K$. This element $a$ specifies a cocycle $c$ by $$ c(t^i,t^j) = \begin{cases} 1 & \text{if } i+j < n, \\ a & \text{if } i+j \ge n. \end{cases} $$

It thus makes sense to denote $A$ simply by $(L,t,a)$. However $a$ is not uniquely specified by $A$ since we can multiply $u$ by any element $\lambda$ of $L^*$ and then $a$ is multiplied by the product of the conjugates of λ. Hence $A$ corresponds to an element of the norm residue group $K^*/N_{L/K}L^*$. We obtain the isomorphisms $$ \mathop{Br}(L/K) \equiv K^*/\mathrm{N}_{L/K} L^* \equiv \mathrm{H}^2(G,L^*) \ . $$

References

• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Universitext. Translated from the German by Silvio Levy. With the collaboration of the translator. Springer-Verlag. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.