Latest revision as of 19:18, 29 January 2018

Eulerian number

A combinatorial counting function for the number of descents in a permutation. Here we take a permutation $(a_1,\ldots,a_n)$ of $(1,\ldots,n)$ and count as a descent any $i$ such that $a_i > a_{i+1}$. We let $$ \left\langle{ n \atop k }\right\rangle $$ denote the number of permutations on $n$ elements with $k$ descents. It satisfies the recurrence relation $$ \left\langle{ n \atop k }\right\rangle = (n-k) \left\langle{ n-1 \atop k-1 }\right\rangle + (k+1) \left\langle{ n-1 \atop k }\right\rangle $$

The Eulerian polynomial is the generating function $$ S_n(t) = \sum_{k=0}^n \left\langle{ n \atop k }\right\rangle t^k \ . $$ The recurrence relation may be written as $$ S_{n+1}(t) = (1+nt) S_n(t) + t(1-t)S'_n(t) \ . $$

The Eulerian numbers appear in a related context. We define an excedance of a permutation to be the number of $i$ such that $a(i) > i$ (weak if $a_i \ge i$). Then the number of permutations with $k$ excendances is equal to the number with $k+1$ weak excedances, and is in turn equal to $\left\langle{ n \atop k }\right\rangle$.

References

T. Kyle Petersen Eulerian Numbers Birkhäuser (2015) ISBN 1-4939-3091-5 Zbl 06467929
Richard P. Stanley Enumerative combinatorics I Wadsworth & Brooks/Cole (1986) ISBN 0-534-06546-5 0608.05001 Zbl 0608.05001

Lattice valuation

A function $\nu$ on a lattice $L$ with values in a ring $R$ satisfying $$ \nu(x \wedge y) + \nu(x \vee y) = \nu(x) + \nu(y) \ . $$

References

Rota, Gian-Carlo (with P. Doubilet, C. Greene, D. Kahaner, A: Odlyzko and R. Stanley) Finite operator calculus Academic Press (1975) ISBN 0-12-596650-4 Zbl 0328.05007

Series-parallel graph

A class of graphs related to ideas from electrical networks. It is convenient to take "graph" to mean unoriented graph allowing loops and multiple edges. A two-terminal series-parallel graph $(G,h,t)$ has two distinguished vertices, source $h$ and sink $t$ (or "head and "tail"). The class is built recursively from the single edge $P_2 = ((\{h,t\}, \{ht\}), h,t)$ with $h$ as head and $t$ as tail, using the operations of series and parallel combination. It is assumed that the graphs to be combined have disjoint vertex sets. The series combination of $(G_1, h_1,t_1)$ and $(G_2, h_2,t_2)$ is the graph obtained by identifying $t_1$ with $h_2$ and taking $h_1$ as head and $t_2$ as tail. The parallel combination of $(G_1, h_1,t_1)$ and $(G_2, h_2,t_2)$ is the graph obtained by identifying $h_1$ with $h_2$ and $t_1$ with $t_2$ then taking $h_1=h_2$ as head and $t_1=t_2$ as tail.

...

Series-parallel graphs are characterised by having no subgraph homeomorphic to $K_4$, the complete graph on $4$ vertices.

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

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 $(x_1,x_2) \mapsto (x_1^*,-x_2)$.

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

How to Cite This Entry:
Richard Pinch/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox&oldid=35996

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Difference between revisions of "User:Richard Pinch/sandbox"

Latest revision as of 19:18, 29 January 2018

Contents

Eulerian number

References

Lattice valuation

References

Series-parallel graph

References

Polarity

References

Frame

References

Alexandrov topology

References

Exponentiation

Positive real numbers

Complex numbers

General algebraic systems

Logarithm

I-semigroup

References

Composition algebra

References

Cayley–Dickson process

Free differential calculus

References

Martin's axiom

References

@@ Line 1: / Line 1: @@
-=Isomorphism theorems=
+=Eulerian number=
-Three theorems relating to [[homomorphism]]s of general [[algebraic system]]s.
+A combinatorial counting function for the number of descents in a permutation.  Here we take a permutation $(a_1,\ldots,a_n)$ of $(1,\ldots,n)$ and count as a ''descent'' any $i$ such that $a_i > a_{i+1}$.  We let
-==First Isomorphism Theorem==
-Let $f : A \rightarrow B$ be a [[homomorphism]] of $\Omega$-algebras and $Q$ the [[Kernel of a function|kernel]] of $f$, as an [[equivalence relation]] on $A$.  Then $Q$ is a [[Congruence (in algebra)|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
 $$
-A_1 / Q_1 \cong A_1^Q / Q \ .
+\left\langle{ n \atop k }\right\rangle
 $$
+denote the number of permutations on $n$ elements with $k$ descents.  It satisfies the [[recurrence relation]]
-==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 a function|kernel]] of $\theta$ on $A/Q$ then there is an isomorphism
 $$
-(A/Q)/(R/Q) \cong A/R \ .
+\left\langle{ n \atop k }\right\rangle = (n-k) \left\langle{ n-1 \atop k-1 }\right\rangle + (k+1) \left\langle{ n-1 \atop k }\right\rangle
 $$
-==References==
+The ''Eulerian polynomial'' is the generating function
-* Paul M. Cohn, ''Universal algebra'', Kluwer (1981) ISBN 90-277-1213-1
-=Lucas sequences=
-Given integers $P$, $Q$ with $D =P^2 - 4Q \neq 0$, the Lucas sequences of the first kind, $U_n = U_n(P,Q)$ and of the second kind $V_n = V_n(P,Q)$ are [[recursive sequence]]s defined by the same linear recurrence relation
-\begin{equation}\label{eq:1}
-X_{n+1} = P.X_n - Q.X_{n-1}
-\end{equation}
-but with starting values $U_0 = 0$, $U_1 = 1$ and $V_0 = 2$, $V_1 = P$.
-The characteristic equation of the recurrence relation is $X^2 = P.X - Q$ with roots $\alpha,\beta = (P \pm \sqrt{D})/2$.  In terms of $\alpha,\beta$ we have
 $$
-U_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}\ ,
+S_n(t) = \sum_{k=0}^n \left\langle{ n \atop k }\right\rangle t^k \ .
 $$
+The recurrence relation may be written as
 $$
-V_n = \alpha^n + \beta^n \ ,
+S_{n+1}(t) = (1+nt) S_n(t) + t(1-t)S'_n(t) \ .
-$$
-so that
-$$
-\alpha^n,\beta^n = \frac{U_n \pm V_n\sqrt{D}}{2} \ .
 $$
-In the case $P=1$,$Q=1$ the [[Fibonacci numbers]] and [[Lucas numbers]] are the Lucas sequences of the first and second kind respectively.
+The Eulerian numbers appear in a related context.  We define an ''excedance'' of a permutation to be the number of $i$ such that $a(i) > i$ (''weak'' if $a_i \ge i$).  Then the number of permutations with $k$ excendances is equal to the number with $k+1$ weak excedances, and is in turn equal to $\left\langle{ n \atop k }\right\rangle$.
-The [[z-transform]] of a Lucas sequence $\sum_{n=0}^\infty X_n z^{-n}$ satisfying \eqref{eq:1} is a rational function with denominator $z^2 - Pz + Q$.  Indeed,
+====References====
-$$
+* T. Kyle Petersen ''Eulerian Numbers'' Birkhäuser (2015)  ISBN 1-4939-3091-5 {{ZBL|06467929}}
-\sum_{n=0}^\infty U_n z^{-n} = \frac{1}{\alpha-\beta}\left(\frac{z}{z-\alpha} - \frac{z}{z-\beta}\right) = \frac{z}{z^2 - Pz + Q} \ .
+* Richard P. Stanley ''Enumerative combinatorics'' '''I''' Wadsworth & Brooks/Cole (1986) ISBN 0-534-06546-5 {{ZBL| 0608.05001}}
-$$
-$$
-\sum_{n=0}^\infty V_n z^{-n} = \frac{z}{z-\alpha} + \frac{z}{z-\beta} = \frac{z(2z-P)}{z^2 - Pz + Q} \ .
-$$
-===Lehmer numbers===
+=Lattice valuation=
-The '''Lehmer numbers''' are a modified form of Lucas sequence of the first kind, defined in terms of $\alpha, \beta$ algebraic numbers with $\alpha\beta$ and $(\alpha+\beta)^2$ integers:
+A function $\nu$ on a [[lattice]] $L$ with values in a [[ring]] $R$ satisfying
 $$
-W_n = \begin{cases}\frac{\alpha^n - \beta^n}{\alpha - \beta},&n\,\text{odd},\\ \frac{\alpha^n - \beta^n}{\alpha^2 - \beta^2},&n\,\text{even}.\end{cases}
+\nu(x \wedge y) + \nu(x \vee y) = \nu(x) + \nu(y) \ .
 $$
-==References==
-* Paulo Ribenboim, "My Numbers, My Friends: Popular Lectures on Number Theory". Springer-Verlag (2002) ISBN 0-387-98911-0.
-* Arthur T. Benjamin; Jennifer J. Quinn, "Proofs that Really Count". Mathematical Association of America (2003) ISBN 0883853337.
-*  Wladyslaw Narkiewicz, "Rational Number Theory in the 20th Century: From PNT to FLT", Springer Monographs in Mathematics, Springer (2011) ISBN 0857295322
-=Dilworth number=
-A numerical invariant of a [[graph]] (cf. [[Graph, numerical characteristics of a]].  The Dilworth number of the graph $G$ is the maximum of vertices in a set $D$ for which the neighbourhoods form a [[Sperner family]]: for any pair of such vertices $x,y \in D$, the neighbourhoods $N(x)$, $N(y)$ each has at least one element not contained in the other.  It is the [[Width of a partially ordered set|width]] of the set of neighbourhoods partially ordered by inclusion.
-The [[diameter]] of a graph exceeds its Dilworth number by at most 1.
+====References====
+* Rota, Gian-Carlo (with P. Doubilet, C. Greene, D. Kahaner, A: Odlyzko and R. Stanley) ''Finite operator calculus'' Academic Press (1975) ISBN 0-12-596650-4 {{ZBL|0328.05007}}
-There is a polynomial time algorithm for computing the Dilworth number of a finite graph.
-==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}}
-=Quaternion algebra=
-An associative algebra over a field that generalises the construction of the [[quaternion]]s over the field of real numbers.
-The quaternion algebra $(a,b)_F$ over a field $F$ is the four-dimensional vector space with basis $\mathbf{1}, \mathbf{i}, \mathbf{j}, \mathbf{k}$ and multiplication defined by
+=Series-parallel graph=
-$$
+A class of [[graph]]s related to ideas from electrical networks.  It is convenient to take "graph" to mean unoriented graph allowing loops and multiple edges.  A two-terminal series-parallel graph $(G,h,t)$ has two distinguished vertices, ''source'' $h$ and ''sink'' $t$ (or "head and "tail").  The class is built recursively from the single edge $P_2 = ((\{h,t\}, \{ht\}), h,t)$ with $h$ as head and $t$ as tail, using the operations of series and parallel combination.  It is assumed that the graphs to be combined have disjoint vertex sets.  The series combination of $(G_1, h_1,t_1)$ and $(G_2, h_2,t_2)$ is the graph obtained by identifying $t_1$ with $h_2$ and taking $h_1$ as head and $t_2$ as tail.  The parallel combination  of $(G_1, h_1,t_1)$ and $(G_2, h_2,t_2)$ is the graph obtained by identifying $h_1$ with $h_2$ and $t_1$ with $t_2$ then taking $h_1=h_2$ as head and $t_1=t_2$ as tail.
-\mathbf{i}^2 = a\mathbf{1},\ \ \mathbf{j}^2 = b\mathbf{1},\ \ \mathbf{i}\mathbf{j} = -\mathbf{j}\mathbf{i} = \mathbf{k}\ .
-$$
-It follows that $\mathbf{k}^2 = -ab\mathbf{1}$ and that any two of $\mathbf{i}, \mathbf{j}, \mathbf{k}$ anti-commute.
-The construction is symmetric: $(a,b)_F = (b,a)_F$.  The algebra so constructed is a [[central simple algebra]] over $F$.
+...
-The algebra $(1,1)_F$ is isomorphic to the $2 \times 2$ matrix ring $M_2(F)$.  Such a quaternion algebra is termed ''split''.
+Series-parallel graphs are characterised by having no subgraph homeomorphic to $K_4$, the [[complete graph]] on $4$ vertices.
 ==References==
-* Tsit-Yuen Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''' , American Mathematical Society (2005) ISBN 0-8218-1095-2 {{ZBL|1068.11023}} {{MR|2104929}}
+* 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}}
-=Square-free=
+=Polarity=
-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 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 over the real or complex numbers, this is equivalent to having no repeated roots.
+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 set]]s, $F : \mathcal{P}A \rightarrow \mathcal{P}B$ and $G : \mathcal{P}B \rightarrow \mathcal{P}A$ by
+$$
+F(U) = \{ b \in B : aRb\ \text{for all}\ a \in U \}
+$$
+and
+$$
+G(V) = \{ a \in A : aRb\ \text{for all}\ b \in V \} \ .
+$$
-An element $x$ of a [[monoid]] $M$ is square-free if the only $y \in M$ such that $y^2$ divides $x$ are units.
+Make $\mathcal{P}A$, $\mathcal{P}B$ [[partially ordered set]]s 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 operator]]s.
-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.
+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
+$$
+\bigvee_{i \in I} (U_i,V_i) = \left({ GF\left({ \bigcup_{i \in I} U_i }\right), \bigcap_{i\in I} V_i  }\right) \ .
+$$
-=Root of unity=
+Every complete lattice $L$ arises in this way: indeed, $L = \mathfrak{B}(L,L,{\le})$.
-{{TEX|done}}
-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 $\zeta$.
+==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}}
-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 $m$.
+=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.
-In the field of [[complex number]]s, there are roots of unity of every order: those of order $m$ take the form
+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
 $$
-\cos \frac{2\pi k}{m} + i \sin   \frac{2\pi k}{m}
+\bigwedge \{ S \} = \mathrm{Int}\left({ \bigcap \{ S \}  }\right)
 $$
-where $0 < k < m$ and $k$ is relatively prime to $m$.
+where $\mathrm{Int}$ denotes the [[interior]].
-====References====
-<table>
-<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1984)</TD></TR>
-</table>
-=Law of quadratic reciprocity=
-==Gauss reciprocity law==
-A relation connecting the values of the Legendre symbols (cf. [[Legendre symbol|Legendre symbol]]) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. [[Quadratic reciprocity law|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 [[#References|[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 residue|cubic]] and [[biquadratic residue]]s led Gauss to introduce the ring of [[Gaussian integer]]s.
 ====References====
 <table>
-<TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,   "Disquisitiones Arithmeticae" , Yale Univ. Press  (1966)  (Translated from Latin)</TD></TR>
+<TR><TD valign="top">[1]</TD> <TD valign="top"> Steven Vickers ''Topology via Logic'' Cambridge Tracts in Theoretical Computer Science '''5''' Cambridge University Press (1989) ISBN 0-521-36062-5 {{ZBL|0668.54001}} </TD></TR>
-<TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. [I.M. Vinogradov] Winogradow,   "Elemente der Zahlentheorie" , R. Oldenbourg  (1956)  (Translated from Russian)</TD></TR>
+<TR><TD valign="top">[2]</TD> <TD valign="top"> Jonathan S. Golan ''Semirings and their Applications'' Springer (2013)  ISBN 9401593337</TD></TR>
-<TR><TD valign="top">[3]</TD> <TD valign="top">  H. Hasse,   "Vorlesungen über Zahlentheorie" , Springer  (1950)</TD></TR>
 </table>
+=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.
-====Comments====
+The Alexandrov topology makes $X$ a [[T0 space]] and the [[Specialization of a point|specialisation]] order is just the original order ${\le}$ on $X$.
-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|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|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|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====
-<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Z.I. Borevich,   I.R. Shafarevich,   "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR></table>
+* 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.
-====Comments====
+The repeated operations may be carried out in any order provided that multiplication is [[associativity|associative]], $x \cdot (y \cdot z) = (x \cdot y) \cdot z$.  In this case we have
-See also [[Quadratic residue|Quadratic residue]]; [[Dirichlet character|Dirichlet character]].
+$$
+x^{m+n} = x^m \cdot x^n \ .
-====References====
+$$
-<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. XIII</TD></TR></table>
-=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. [[#References|[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 [[#References|[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 $x < 10^{1000000}$, [[#References|[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  [[#References|[c2]]].  For some results on this still (1996) largely open problem, see [[#References|[c3]]] and the references therein.
-====References====
-<table>
-<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Chandrasekharan,   "Introduction to analytic number theory" , Springer  (1968) {{MR|0249348}} {{ZBL|0169.37502}}</TD></TR>
-<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}}</TD></TR>
-<TR><TD valign="top">[c1]</TD> <TD valign="top">  A. Schlafly,   S. Wagon,   "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$"  ''Math. Comp.'' , '''63'''  (1994)  pp. 415–419</TD></TR>
-<TR><TD valign="top">[c2]</TD> <TD valign="top">  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}}</TD></TR>
-<TR><TD valign="top">[c3]</TD> <TD valign="top">  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</TD></TR>
-<TR><TD valign="top">[c4]</TD> <TD valign="top">  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</TD></TR>
-<TR><TD valign="top">[c5]</TD> <TD valign="top">  R. Sivamarakrishnan,   "The many facets of Euler's totient II: generalizations and analogues"  ''Nieuw Archief Wiskunde (4)'' , '''8'''  (1990)  pp. 169–188</TD></TR>
-<TR><TD valign="top">[c6]</TD> <TD valign="top">  R. Sivamarakrishnan,   "The many facets of Euler's totient I"  ''Nieuw Archief Wiskunde (4)'' , '''4'''  (1986)  pp. 175–190</TD></TR>
-<TR><TD valign="top">[c7]</TD> <TD valign="top">  L.E. Dickson,   "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint  (1971)  pp. Chapt. V; 113–155</TD></TR>
-</table>
-==Totient function==
+If the operation is also [[commutativity|commutative]] then we have
-''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 function]]s.
-One has
 $$
-J_k(n) = n^k \prod_{p|n} \left({ 1 - p^{-k} }\right)
+(x \cdot y)^n = x^n \cdot y^n \ .
 $$
-where $p$ runs over the prime numbers dividing $n$, and
+We may extend the definition to non-positive integer powers by defining $x^0 = 1$ and
 $$
-J_k(n) = \sum_{d | n} \mu(n/d) d^k
+x^{-n} = \frac{1}{x^n}
 $$
-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.
+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 $q < 0$.
-====References====
+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.
-<table>
-<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Schlafly,   S. Wagon,   "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$"  ''Math. Comp.'' , '''63'''  (1994)  pp. 415–419</TD></TR>
-<TR><TD valign="top">[a2]</TD> <TD valign="top">  D.H. Lehmer,   "On Euler's totient function"  ''Bull. Amer. Math. Soc.'' , '''38'''  (1932)  pp. 745–751</TD></TR>
-<TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR>
-<TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR>
-<TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Sivamarakrishnan,   "The many facets of Euler's totient II: generalizations and analogues"  ''Nieuw Archief Wiskunde (4)'' , '''8'''  (1990)  pp. 169–188</TD></TR>
-<TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Sivamarakrishnan,   "The many facets of Euler's totient I"  ''Nieuw Archief Wiskunde (4)'' , '''4'''  (1986)  pp. 175–190</TD></TR>
-<TR><TD valign="top">[a7]</TD> <TD valign="top">  L.E. Dickson,   "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint  (1971)  pp. Chapt. V; 113–155</TD></TR>
-</table>
-==Jordan totient function==
+For positive real numbers there are mutually inverse [[exponential function, real|exponential]] and [[logarithmic function|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-number|$e$]] is the base of natural logarithms.
-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 numbers|coprime]] $(k + 1)$-tuple together with $n$.  This is a generalisation of Euler's [[totient function]], which is $J_1$.
+===Complex numbers===
+Exponentiation of complex numbers with non-integer exponents may be defined using the complex [[exponential function]] and [[logarithmic function|logarithm]].  The exponential function is analytic and defined on the whole complex plane: the logarithmic function requires a choice of [[Branch of an analytic function|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 $z^w = \exp(w \log z)$.
-Jordan's totient function is [[Multiplicative arithmetic function|multiplicative]] and may be evaluated as
+===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
+$$
+x^{n+1} = x \star (x^n)
 $$
-J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) \ .
+and
 $$
+x^{n+1} = (x^n) \star x
-By [[Möbius inversion]] we have $\sum_{d | n } J_k(d) = n^k $.  The [[Average order of an arithmetic function|average order]] of $J_k(n)$ is $c n^k$ for some $c$.
-The analogue of Lehmer's problem for the Jordan totient function (and $k>1$) is easy, [[#References|[c4]]]: For $k>1$, $J_k(n) | n^k-1$ if and only if $n$ is a prime number. Moreover, if $n$ is a prime number, then $J_k(n) = n^k-1$.
-For much more information on the Euler totient function, the Jordan totient function and various other generalizations, see [[#References|[c5]]], [[#References|[c6]]].
-====References====
-* Dickson, L.E. ''History of the Theory of Numbers I'', Chelsea (1971) p. 147,  ISBN 0-8284-0086-5
-* Ram Murty, M. ''Problems in Analytic Number Theory'', Graduate Texts in Mathematics '''206''' Springer-Verlag (2001) p. 11  ISBN 0387951431 {{ZBL|0971.11001}}
-* Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.32–36. ISBN 1-4020-2546-7. {{ZBL|1079.11001}}
-=Multiplicative sequence=
-Also ''m''-sequence, a sequence of [[polynomial]]s associated with a formal group structure.  They have application in the [[cobordism|cobordism ring]] in [[algebraic topology]].
-==Definition==
-Let $K_n$ be polynomials over a ring $A$ in indeterminates $p_1,\ldots$ weighted so that $p_i$ has weight $i$ (with $p_0=1$) and all the terms in $K_n$ have weight $n$ (so that $K_n$ is a polynomial in $p_1,\ldots,p_n$).  The sequence $K_n$ is ''multiplicative'' if an identity
-$$\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 $$
-is the ''characteristic power series'' of the $K_n$.  A multiplicative sequence is determined by its characteristic power series $Q(z)$, and every power series with constant term 1 gives rise to a multiplicative sequence.
-To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of ''z''<sup>''j''</sup> in the product
-$$ \prod_{i=1}^m Q(\beta_i z) $$
-for any $m>j$.  This is symmetric in the $\beta_i$ and homogeneous of weight ''j'': so can be expressed as a polynomial $K_j(p_1,\ldots,p_j)$ in the [[elementary symmetric function]]s $p$ of the $\beta$.  Then $K_j$ defines a multiplicative sequence.
-==Examples==
-As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$.
-Consider the power series
-$$ Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k
 $$
-where $B_k$ is the $k$-th [[Bernoulli number]].  The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$.
+respectively.  A binary operation is [[power associativity|power associative]] if the powers of a single element form an associative subsystem, so that exponentiation is well-defined.
-The multiplicative sequence with characteristic power series
+=Logarithm=
+The operation inverse to [[exponentiation]].
-$$ Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} $$
+Over the fields of [[Real number|real]] or [[complex number]]s, one speaks of the [[logarithm of a number]].  The [[logarithmic function]] is the complex analytic function inverse to the [[exponential function]].
-is denoted $A_j(p_1,\ldots,p_j)$.
+In a finite Abelian group, the [[discrete logarithm]] is the inverse to exponentiation, with applications in [[cryptography]].
-The multiplicative sequence with characteristic power series
+The [[Zech logarithm]] in a finite field is related to the discrete logarithm.
-$$Q(z) = \frac{z}{1-\exp(-z)}  = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} $$
-is denoted $T_j(p_1,\ldots,p_j)$: the ''[[Todd polynomial]]s''.
-==Genus==
-The '''genus''' of a multiplicative sequence is  a [[ring homomorphism]], from the  [[cobordism|cobordism ring]] of smooth oriented [[compact manifold]]s to another [[ring]], usually the ring of [[rational number]]s.
-For example, the [[Todd genus]] is associated to the Todd polynomials $T_j$ with characteristic power series $$\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} . $$
+=''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.
-==References==
+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)$.
-* 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 matrix|invertible matrices]] over the [[ring of polynomials]] over a [[field]].  It has been extended by [[Jean-Pierre Serre|Serre]] to give a description of the structure of the corresponding matrix group over the [[coordinate ring]] of a [[projective curve]].
-==Nagao's theorem==
+====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}}
-For a [[Ring (mathematics)|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
+=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 number]]s are the real numbers, the field of [[complex number]]s $\mathbf{C}$, the [[skew-field]] of [[quaternion]]s, the non-associative algebra of [[octonions]].
-$$ 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 \ . $$
+====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}}
-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 [[Singular point of an algebraic variety|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. {{MR|0114866}}.  {{ZBL|0092.02504}}.
-* Serre, Jean-Pierre. ''Trees''. (Springer, 2003) ISBN 3-540-44237-5.
-=Erdős–Wintner theorem=
-A result in [[probabilistic number theory]] characterising those [[additive function]]s that possess a limiting distribution.
-==Limiting distribution==
-A distribution function $F$ is a non-decreasing function from the real numbers to the unit interval [0,1] which is right-continuous and has limits 0  at $-\infty$ and 1 at $+\infty$.
-Let $f$ be a complex-valued function on natural numbers.  We say that $F$ is a limiting distribution for $f$ if $F$ is a distribution function and the sequence $F_N$ defined by
+=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 number]]s, [[quaternion]]s 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
 $$
-F_n(t) = \frac{1}{N} | \{n \le N : |f(n)| \le t \} |
+(a_1,a_2) \cdot (b_1,b_2) = (a_1b_1 - d b_2 a_2^* , a_1^*b_2 + b_1a_2) \ .
 $$
+The algebra $A_1$ has an involution $(x_1,x_2) \mapsto (x_1^*,-x_2)$.
-[[Weak convergence of probability measures|converges weakly]] to $F$.
-==Statement of the theorem==
+=Free differential calculus=
-Let $f$ be an additive function.  There is a limiting distribution for $f$ if and only if the following three series converge:
+Let $F$ be a [[free group]] on a set of generators $X = \{x_i : i \in I \}$ and $R[F]$ the [[group ring]] of $F$ over a commutative unital ring $R$.  The ''Fox derivative'' $\partial_i$ are maps from $F$ to $R[F]$ defined by
 $$
-\sum_{|f(p)|>1} \frac{1}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)^2}{p} \ .
+\partial_i(x_j) = \delta_{ij} \ ,
 $$
-When these conditions are satisfied, the distribution is given by
-$$
-F(t) = \prod_p \left({1 - \frac{1}{p} }\right) \cdot \left({1 + \sum_{k=1}^\infty p^{-k}\exp(i t f(p)^k) }\right) \ .
-$$
-==References==
-* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. ''Handbook of number theory I''. Dordrecht: Springer-Verlag (2006). pp. 564–566. ISBN 1-4020-4215-9.  {{ZBL|1151.11300}}
-* Tenenbaum, Gérald ''Introduction to Analytic and Probabilistic Number Theory''. Cambridge studies in advanced mathematics '''46'''. Cambridge University Press (1995). ISBN 0-521-41261-7.  {{ZBL|0831.11001}}
-=Brauer–Wall group=
-A [[group]] classifying graded [[central simple algebra]]s over a field.  It was first defined by Wall (1964) as a generalisation of the [[Brauer group]].
-The Brauer group $\mathrm{B}(F)$ of a field $F$ is defined on the isomorphism classes of central simple algebras over ''F''.  The analogous construction for $\mathbf{Z}/2$-[[graded algebra]]s defines the Brauer–Wall group $\mathrm{BW}(F)$.{{cite|Lam (2005) pp.98–99}}
-==Properties==
-* The Brauer group $\mathrm{B}(F)$ injects into $\mathrm{BW}(F)$ by mapping a CSA $A$ to the graded algebra which is $A$ in grade zero.
-There is an exact sequence
-$$
-\rightarrow \mathrm{B}(F) \rightarrow \mathrm{BW}(F) \rightarrow Q(F) \rightarrow 0
-$$
-where $Q(F)$ is the group of graded quadratic extensions of $F$, defined as $\mathbf{Z}/2 \times F^*/(F^*)^2$ with multiplication $(e,x)(f,y) = (e+f,(-1)^{ef}xy$. The map from W to BW is the '''[[Clifford invariant]]''' defined by mapping an algebra to the pair consisting of its grade and [[Determinant of a quadratic form|determinant]].
-There is a map from the additive group of the [[Witt–Grothendieck ring]] to the Brauer–Wall group obtained by sending a quadratic space to its [[Clifford algebra]].  The map factors through the [[Witt group]]{{cite|Lam (2005) p.113}}  which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.{{cite|Lam (2005) p.115}}
-==Examples==
-* $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$.  This is an algebraic aspect of [[Bott periodicity]].
-==References==
-* Lam, Tsit-Yuen, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', (American Mathematical Society, 2005) ISBN 0-8218-1095-2 {{MR|2104929}}, {{ZBL|1068.11023}}
-* Wall, C. T. C., "Graded Brauer groups", ''Journal für die reine und angewandte Mathematik'' '''213''' (1964) 187–199, ISSN 0075-4102, {{ZBL|0125.01904}}, {{MR|0167498}}
-=Factor system=
-A function on a [[group]] giving the data required to construct an [[algebra]].  A factor system constitutes a realisation of the cocycles in the second [[cohomology group]] in [[group cohomology]].
-Let $G$ be a group and $L$ a field on which $G$ acts as automorphisms.  A ''cocycle'' or ''factor system'' is a map $c : G \times G \rightarrow L^*$ satisfying
 $$
-c(h,k)^g c(hk,g) = c(h,kg) c(k,g) \ .
+\partial_i(1) = 0 \ ,
 $$
-Cocycles are ''equivalent'' if there exists some system of elements $a : G \rightarrow L^*$ with
 $$
-c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) \ .
+\partial_i(uv) = u \partial_i(v) + \partial_i(u) v \ .
 $$
+It follows that
-Cocycles of the form
 $$
-c(g,h) = a_g^h a_h a_{gh}^{-1}
+\partial_i(x_i^{-1}) = - x_i^{-1} \ .
 $$
-are called ''split''.  Cocycles under multiplication modulo split cocycles form a group, the second cohomology group $H^2(G,L^*)$.
-==Crossed product algebras==
+The maps extend to [[derivation]]s on $R[F]$.
-Let us take the case that $G$ is the [[Galois group]] of a [[field extension]] $L/K$.  A factor system $c$ in $H^2(G,L^*)$ gives rise to a ''crossed product algebra'' $A$, which is a $K$-algebra containing $L$ as a subfield, generated by the elements $\lambda \in L$ and $u_g$ with multiplication
-$$
-\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) \ .$$
-Every [[central simple algebra]] over$K$ that splits over $L$ arises in this way.  The tensor product of algebras corresponds to multiplication of the corresponding elements in$H^2$.  We thus obtain an identification of the [[Brauer group]], where the elements are classes of CSAs over $K$, with $H^2$.{{cite|Saltman (1999) p.44}}
+===References===
+* D. L. Johnson, ''Presentations of Groups'', London Mathematical Society Student Texts '''15''' Cambridge University Press (1997) ISBN 0-521-58542-2
-==Cyclic algebra==
+=Martin's axiom=
-Let us further restrict to the case that $L/K$ is [[Cyclic extension|cyclic]] with Galois group $G$ of order $n$ generated by $t$.  Let $A$ be a crossed product $(L,G,c)$ with factor set $c$.  Let $u=u_t$ be the generator in $A$ corresponding to $t$.  We can define the other generators
+An axiom of set theory.  Let $(P,{<})$ be a [[partially ordered set]] satisfying the [[countable chain condition]] and $D$ a family of $\mathfrak{k}$ dense subsets of $P$ for $\mathfrak{k}$ a cardinal less than $2^{\aleph_0}$.  Then $\text{MA}_{\mathfrak{k}}$ asserts that there is a $D$-generic filter on $P$.  Martin's axiom $\text{MA}$ is the conjunction of $\text{MA}_{\mathfrak{k}}$ for all $\mathfrak{k} < 2^{\aleph_0}$.
-$$
-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}
-$$
-It thus makes sense to denote $A$ simply by $(L,t,a)$.  However $a$ is not uniquely specified by $A$ since we can multiply $u$ by any element $\lambda$ of $L^*$ and then $a$ is multiplied by the product of the conjugates of λ.  Hence $A$ corresponds to an element of the norm residue group
+The case $\text{MA}_{\aleph_0}$ holds in [[ZFC]].  MA is a consequence of the [[Continuum hypothesis]] ($\text{CH}$) but $\text{MA} \wedge \text{CH}$ is consistent with ZFC if ZFC is consistent.
-$(K^*/N_{L/K}L^*$.  We obtain the isomorphisms
-$$
-\mathop{Br}(L/K) \equiv K^*/\mathrm{N}_{L/K} L^* \equiv \mathrm{H}^2(G,L^*) \ .
-$$
-==References==
+====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.
+* Thomas Jech, ''Set Theory'', Perspectives in Mathematical Logic, Third Millennium Edition, revised and expanded.  Springer (2007) ISBN 3-540-44761-X
-* 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.