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=35555

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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: @@
-=Sum of divisors function=
+=Eulerian number=
-''of a natural number $n$''
+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
-The sum of the positive integers divisors of a natural number $n$, including $1$ and $n$:
 $$
-\sigma(n) = \sum_{d | n} d \ .
+\left\langle{ n \atop k }\right\rangle
 $$
-More generally, the function $\sigma_k$ is defined as
+denote the number of permutations on $n$ elements with $k$ descents.  It satisfies the [[recurrence relation]]
 $$
-\sigma_k(n) = \sum_{d | n} d^k \ .
+\left\langle{ n \atop k }\right\rangle = (n-k) \left\langle{ n-1 \atop k-1 }\right\rangle + (k+1) \left\langle{ n-1 \atop k }\right\rangle
 $$
-so that $\sigma = \sigma_1$ and the number of divisors function $\tau = \sigma_0$.
-These are [[multiplicative arithmetic function]]s with [[Dirichlet series]]
+The ''Eulerian polynomial'' is the generating function
 $$
-\sum_{n=1}^\infty \sigma_k(n) n^{-s} = \prod_p \left({(1-p^{-s})(1-p^{k-s}) }\right)^{-1} = \zeta(s) \zeta(s-k)\ .
+S_n(t) = \sum_{k=0}^n \left\langle{ n \atop k }\right\rangle t^k \ .
+$$
+The recurrence relation may be written as
+$$
+S_{n+1}(t) = (1+nt) S_n(t) + t(1-t)S'_n(t) \ .
 $$
-The average order of $\sigma(n)$ is given by
+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 {{ZBL| 0608.05001}}
+=Lattice valuation=
+A function $\nu$ on a [[lattice]] $L$ with values in a [[ring]] $R$ satisfying
 $$
-\sum_{n \le x} \sigma(n) = \frac{\pi^2}{12} x^2 + O(x \log x) \ .
+\nu(x \wedge y) + \nu(x \vee y) = \nu(x) + \nu(y) \ .
 $$
-There are a number of well-known classes of number characterised by their divisor sums.
-A ''[[perfect number]]'' $n$ is the sum of its [[aliquot divisor]]s (those divisors other than $n$ itself), so $\sigma(n) = 2n$.  The even perfect numbers are characterised in terms of [[Mersenne prime]]s $P = 2^p-1$ as $n = 2^{p-1}.P$: it is not known if there are any odd perfect numbers.  An ''[[almost perfect number]]'' $n$ similarly has the property that $\sigma(n) = 2n-1$: these include the powers of 2.  A ''quasiperfect number'' is defined by $\sigma(n) = 2n+1$: it is known known if any exists.  See also [[Descartes number]].
 ====References====
-* Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation '''36''' (1981) 583–586. ISSN 0025-5718.  {{ZBL|0472.10007}}
+* 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}}
-* G. Tenenbaum, ''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}}
-=Exponential law=
-A term with a variety of meanings.
-A process which changes by a given factor over a fixed time period.
-A property of the [[exponential function]], such as $\exp(x+y) = \exp(x)\exp(y)$ or of a [[power]], such as $x^{yz} = (x^y)^z$.
+=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.
-A property of sets of maps between sets, such as $\mathrm{Map}(X\times Y,Z) \leftrightarrow \mathrm{Map}(X, \mathrm{Map}(Y,Z))$, see [[Exponential law for sets]].
+...
-A property of sets of continuous maps between topological spaces, see [[Exponential law (in topology)]].
+Series-parallel graphs are characterised by having no subgraph homeomorphic to $K_4$, the [[complete graph]] on $4$ vertices.
-=Exponential law for sets=
+==References==
-The correspondence between the sets $A^{B \times C}$ and $(A^B)^C$, where $X^Y$ denotes the set of all maps from the set $Y$ to the set $X$.  Given $f \in A^{B \times C}$, that is $F : B \times C \rightarrow A$, and given $c \in C$, let $f_c$ denote the map $f_c : B \rightarrow A$ by $f_c : b \mapsto f(b,c)$.  Then $c \mapsto f_c$ defines a map from $A^{B \times C} \rightarrow (A^B)^C$.  In the opposite direction, let $G \in (A^B)^C$.  Given $b \in B$ and $c \in C$, define $g(b,c)$ to be $G(c)$ applied to $b$.  Then $G \mapsto g$ defines a map from $(A^B)^C \rightarrow A^{B \times C}$.  These two correspondences are mutually inverse.
+* 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}}
-In computer science, this operation is known as "Currying" after Haskell Curry (1900-1982).
-In category-theoretic terms, the exponential law makes the [[category of sets]] a [[Cartesian-closed category]].
+=Polarity=
-In [[cardinal arithmetic]], this corresponds to the formula
+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
 $$
-\mathfrak{a}^{\mathfrak{b}\mathfrak{c}} = (\mathfrak{a}^{\mathfrak{b}})^{\mathfrak{c}} \ .
+F(U) = \{ b \in B : aRb\ \text{for all}\ a \in U \}
 $$
+and
-There are further correspondences relating the sets $A^{B \coprod C}$ and $A^B \times A^C$ and the sets $(A \times B)^C$ and $A^C \times B^C$, corresponding to the formulae in cardinal arithmetic $\mathfrak{a}^{\mathfrak{b} + \mathfrak{c}} = \mathfrak{a}^{\mathfrak{b}}\mathfrak{a}^{\mathfrak{c}}$ and $(\mathfrak{a} \mathfrak{b})^{\mathfrak{c}} = \mathfrak{a}^{\mathfrak{c}} \mathfrak{a}^{\mathfrak{c}}$.
-====References====
-* Benjamin C. Pierce, ''Basic Category Theory for Computer Scientists'', MIT Press (1991) ISBN 0262660717
-* Paul Taylor, ''Practical Foundations of Mathematics'', Cambridge Studies in Advanced Mathematics '''59''', Cambridge University Press (1999) ISBN 0-521-63107-6
-=Matrix multiplication=
-A [[binary operation]] on compatible [[matrix|matrices]] over a [[ring]] $R$.  There are several such operations.
-===Cayley multiplication===
-Most usually what is referred to as "matrix multiplication".  The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $AB$ with entries
 $$
-(AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}\ ,\ \ i=1,\ldots,m\,\ j=1,\ldots,p\,.
+G(V) = \{ a \in A : aRb\ \text{for all}\ b \in V \} \ .
 $$
-The multiplication corresponds to composition of linear maps.  If $A$ is the matrix of a linear map $\alpha : R^m \rightarrow R^n$ and $B$ is the matrix of a linear map $\beta : R^n \rightarrow R^p$, then $AB$ is the matrix of the linear map $\alpha\beta : R^m \rightarrow R^p$.
+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.
-===Hadamard multiplication===
+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
-The Hadamard product, or Schur product, of two $m \times n$ matrices $A$ and  $B$ is the $m \times n$ matrix $AB$ with
 $$
-(A \circ B)_{ij} = a_{ij} b_{ij}\ ,\ \ i=1,\ldots,m\,\ j=1,\ldots,n\,.
+\bigwedge_{i \in I} (U_i,V_i) = \left({ \bigcap_{i\in I} U_i, FG\left({ \bigcup_{i \in I} V_i }\right) }\right)
 $$
+and
-===Kronecker multiplication===
-The Kronecker product, also tensor product or direct product, of an $m \times n$ matrix $A$ and an $p \times q$ matrix $B$ is the $mp \times nq$ matrix $AB$ with entries
 $$
-(A \otimes B)_{(i-1)p+k,(j-1)q+l} = a_{ij} b_{kl},\ \ i=1,\ldots,m\,\ j=1,\ldots,n\,\ k=1,\ldots,p\,\ l=1,\ldots,q\,.
+\bigvee_{i \in I} (U_i,V_i) = \left({ GF\left({ \bigcup_{i \in I} U_i }\right), \bigcap_{i\in I} V_i  }\right) \ .
 $$
-====References====
+Every complete lattice $L$ arises in this way: indeed, $L = \mathfrak{B}(L,L,{\le})$.
-* Gene H. Golub, Charles F. Van Loan, ''Matrix Computations'', Johns Hopkins Studies in the Mathematical Sciences '''3''', JHU Press (2013) ISBN 1421407949
-* James E. Gentle, ''Matrix Algebra: Theory, Computations, and Applications in Statistics'', Springer Texts in Statistics, Springer (2007) ISBN 0-387-70872-3
-* Manfred Schroeder, ''Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity'', Springer (2008) ISBN 3-540-85297-2
-=Multiplicative sequence=
+==References==
-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]].
+* 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}}
-==Definition==
+=Frame=
-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
+A generalisation of the concept of topological space occurring in the theory of logic and computation.
-$$\sum_i p_i z^i = \sum p'_i z^i \cdot \sum_i p''_i z^i $$
+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:
+$$
+x \wedge \bigvee \{ y \in Y \} = \bigvee \{ x \wedge y : y \in Y \} \ .
+$$
-implies
+The powerset $\mathcal{P}(A)$ of a set $A$ forms a frame.
-$$\sum_i K_i(p_1,\ldots,p_i) z^i = \sum_j K_j(p'_1,\ldots,p'_j) z^j \cdot \sum_k K_k(p''_1,\ldots,p''_k) z^k  . $$
+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
-The power series
+$$
+\bigwedge \{ S \} = \mathrm{Int}\left({ \bigcap \{ S \}  }\right)
+$$
+where $\mathrm{Int}$ denotes the [[interior]].
-$$\sum K_n(1,0,\ldots,0) z^n $$
+====References====
+<table>
+<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"> Jonathan S. Golan ''Semirings and their Applications'' Springer (2013)  ISBN 9401593337</TD></TR>
+</table>
-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.
+=Alexandrov topology=
+''on a [[partially ordered set]] $(X,{\le})$''
-To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of ''z''<sup>''j''</sup> in the product
+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.
-$$ \prod_{i=1}^m Q(\beta_i z) $$
+The Alexandrov topology makes $X$ a [[T0 space]] and the [[Specialization of a point|specialisation]] order is just the original order ${\le}$ on $X$.
-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.
+====References====
+* Johnstone, Peter T. ''Stone spaces'' Cambridge Studies in Advanced Mathematics '''3''' Cambridge University Press(1986) {{ZBL|0586.54001}}
-==Examples==
+=Exponentiation=
-As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$.
+The algebraic and analytic operations generalising the operation of repeated multiplication in number systems.
-Consider the power series
+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.
-$$ Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k
+The repeated operations may be carried out in any order provided that multiplication is [[associativity|associative]], $x \cdot (y \cdot z) = (x \cdot y) \cdot z$.  In this case we have
+$$
+x^{m+n} = x^m \cdot x^n \ .
 $$
-where $B_k$ is the $k$-th [[Bernoulli number]].  The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$.
-The multiplicative sequence with characteristic power series
+If the operation is also [[commutativity|commutative]] then we have
+$$
+(x \cdot y)^n = x^n \cdot y^n \ .
+$$
-$$ Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} $$
+We may extend the definition to non-positive integer powers by defining $x^0 = 1$ and
+$$
+x^{-n} = \frac{1}{x^n}
+$$
+whenever this makes sense.
-is denoted $A_j(p_1,\ldots,p_j)$.
+We may extend the definition to rational number exponents by taking $x^{1/n}$ to be any number $y$ such that $y^n = x$: this may denote none, one or more than one number.
-The multiplicative sequence with characteristic power series
+===Positive real numbers===
+Exponentiation of positive real numbers by rational exponents may be defined by taking $x^{1/n}$ to be the unique positive real solution of $y^n = x$: this always exists.  We thus have $x^q$ well-defined for $x>0$ and any rational exponent $q$.  Exponentiation preserves order: if $x > y$ then $x^q > y^q$ if $q > 0$ and $x^q < y^q$ if $q < 0$.
-$$Q(z) = \frac{z}{1-\exp(-z)}  = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} $$
+We can now define exponentiation with real exponent $r$ by defining $x^r$ to be the limit of $x^{q_n}$ where $q_n$ is a sequence of rational numbers converging to $r$.  There is always such a sequence, and the limit exists and does not depend on the chosen sequence.
-is denoted $T_j(p_1,\ldots,p_j)$: the ''[[Todd polynomial]]s''.
-==Genus==
+For positive real numbers there are mutually inverse [[exponential function, real|exponential]] and [[logarithmic function|logarithm]] functions which allow the alternative definition
-The '''genus''' of a multiplicative sequence is  a [[ring homomorphism]], from the  [[cobordism|cobordism ring]] of smooth oriented [[compact manifold]]s to another [[ring]], usually the ring of [[rational number]]s.
+$$
+x^y = \exp(y \log x) \ .
+$$
+Here the exponential function may be regarded as $\exp(x) = e^x$ where [[E-number|$e$]] is the base of natural logarithms.
-For example, the [[Todd genus]] is associated to the Todd polynomials $T_j$ with characteristic power series $$\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} . $$
+===Complex numbers===
+Exponentiation of complex numbers with non-integer exponents may be defined using the complex [[exponential function]] and [[logarithmic function|logarithm]].  The exponential function is analytic and defined on the whole complex plane: the logarithmic function requires a choice of [[Branch of an analytic function|branch]], corresponding to a choice of the range of values for the [[argument]], to make it single-valued.  Given such a choice, exponentiation may be defined as $z^w = \exp(w \log z)$.
-==References==
+===General algebraic systems===
-* Hirzebruch, Friedrich. ''Topological methods in algebraic geometry'',  Classics in Mathematics. Translation from the German and appendix one  by R. L. E. Schwarzenberger. Appendix two by A. Borel.  Reprint of the  2nd, corr. print. of the 3rd ed. [1978]  (Berlin: Springer-Verlag, 1995). ISBN 3-540-58663-6. {{ZBL|0843.14009}}.
+For a general (not necessarily associative) binary operation, it is necessary to define the order of operations.  The left and right ''principal powers'' are defined inductively by
+$$
+x^{n+1} = x \star (x^n)
+$$
+and
+$$
+x^{n+1} = (x^n) \star x
+$$
+respectively.  A binary operation is [[power associativity|power associative]] if the powers of a single element form an associative subsystem, so that exponentiation is well-defined.
-=Nagao's theorem=
+=Logarithm=
-A result, named after Hirosi Nagao, about the structure of the [[group]] of 2-by-2 [[Invertible matrix|invertible matrices]] over the [[ring of polynomials]] over a [[field]].  It has been extended by [[Jean-Pierre Serre|Serre]] to give a description of the structure of the corresponding matrix group over the [[coordinate ring]] of a [[projective curve]].
+The operation inverse to [[exponentiation]].
-==Nagao's theorem==
+Over the fields of [[Real number|real]] or [[complex number]]s, one speaks of the [[logarithm of a number]].  The [[logarithmic function]] is the complex analytic function inverse to the [[exponential function]].
-For a [[Ring (mathematics)|general ring]] $R$ we let $GL_2(R)$ denote the group of invertible 2-by-2 matrices with entries in $R$, and let $R^*$ denote the [[group of units]] of $R$, and let
+In a finite Abelian group, the [[discrete logarithm]] is the inverse to exponentiation, with applications in [[cryptography]].
-$$ 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 \ . $$
+The [[Zech logarithm]] in a finite field is related to the discrete logarithm.
-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==
+=''I''-semigroup=
-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)$.
+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)$.
-* 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==
+====References====
-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$.
+* 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}}
-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
+=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]].
+====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}}
-$$
-F_n(t) = \frac{1}{N} | \{n \le N : |f(n)| \le t \} |
-$$
-[[Weak convergence of probability measures|converges weakly]] to $F$.
-==Statement of the theorem==
-Let $f$ be an additive function.  There is a limiting distribution for $f$ if and only if the following three series converge:
-$$
-\sum_{|f(p)|>1} \frac{1}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)^2}{p} \ .
-$$
-When these conditions are satisfied, the distribution is given by
+=Cayley–Dickson process=
+A construction of an algebra $A_1$ from an algebra $A$ with involution over a field $K$ which generalises the construction of the [[complex number]]s, [[quaternion]]s and [[octonion]].  Fix a parameter $d \in A$.  As a set $A_1 = A \times A$ with addition defined by $(a_1,a_2) + (b_1,b_2) = (a_1+b_1, a_2+b_2)$ and multiplication by
 $$
-F(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) \ .
+(a_1,a_2) \cdot (b_1,b_2) = (a_1b_1 - d b_2 a_2^* , a_1^*b_2 + b_1a_2) \ .
 $$
+The algebra $A_1$ has an involution $(x_1,x_2) \mapsto (x_1^*,-x_2)$.
-==References==
-* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. ''Handbook of number theory I''. Dordrecht: Springer-Verlag (2006). pp. 564–566. ISBN 1-4020-4215-9.  {{ZBL|1151.11300}}
-* Tenenbaum, Gérald ''Introduction to Analytic and Probabilistic Number Theory''. Cambridge studies in advanced mathematics '''46'''. Cambridge University Press (1995). ISBN 0-521-41261-7.  {{ZBL|0831.11001}}
-=Brauer–Wall group=
-A [[group]] classifying graded [[central simple algebra]]s over a field.  It was first defined by Wall (1964) as a generalisation of the [[Brauer group]].
-The Brauer group $\mathrm{B}(F)$ of a field $F$ is defined on the isomorphism classes of central simple algebras over ''F''.  The analogous construction for $\mathbf{Z}/2$-[[graded algebra]]s defines the Brauer–Wall group $\mathrm{BW}(F)$.{{cite|Lam (2005) pp.98–99}}
-==Properties==
-* The Brauer group $\mathrm{B}(F)$ injects into $\mathrm{BW}(F)$ by mapping a CSA $A$ to the graded algebra which is $A$ in grade zero.
-There is an exact sequence
+=Free differential calculus=
+Let $F$ be a [[free group]] on a set of generators $X = \{x_i : i \in I \}$ and $R[F]$ the [[group ring]] of $F$ over a commutative unital ring $R$.  The ''Fox derivative'' $\partial_i$ are maps from $F$ to $R[F]$ defined by
 $$
-\rightarrow \mathrm{B}(F) \rightarrow \mathrm{BW}(F) \rightarrow Q(F) \rightarrow 0
+\partial_i(x_j) = \delta_{ij} \ ,
 $$
-where $Q(F)$ is the group of graded quadratic extensions of $F$, defined as $\mathbf{Z}/2 \times F^*/(F^*)^2$ with multiplication $(e,x)(f,y) = (e+f,(-1)^{ef}xy$. The map from W to BW is the '''[[Clifford invariant]]''' defined by mapping an algebra to the pair consisting of its grade and [[Determinant of a quadratic form|determinant]].
-There is a map from the additive group of the [[Witt–Grothendieck ring]] to the Brauer–Wall group obtained by sending a quadratic space to its [[Clifford algebra]].  The map factors through the [[Witt group]]{{cite|Lam (2005) p.113}}  which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.{{cite|Lam (2005) p.115}}
-==Examples==
-* $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$.  This is an algebraic aspect of [[Bott periodicity]].
-==References==
-* Lam, Tsit-Yuen, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', (American Mathematical Society, 2005) ISBN 0-8218-1095-2 {{MR|2104929}}, {{ZBL|1068.11023}}
-* Wall, C. T. C., "Graded Brauer groups", ''Journal für die reine und angewandte Mathematik'' '''213''' (1964) 187–199, ISSN 0075-4102, {{ZBL|0125.01904}}, {{MR|0167498}}
-=Factor system=
-A function on a [[group]] giving the data required to construct an [[algebra]].  A factor system constitutes a realisation of the cocycles in the second [[cohomology group]] in [[group cohomology]].
-Let $G$ be a group and $L$ a field on which $G$ acts as automorphisms.  A ''cocycle'' or ''factor system'' is a map $c : G \times G \rightarrow L^*$ satisfying
 $$
-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.