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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