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=Hoeffding inequality=
A simple inequality giving a bound on the probability of deviation of a sum of independent random variables from a given value.
Let $X_1,\ldots,X_n$ be independent [[Subgaussian distribution|sub-Gaussian random variable]]s with means $\mu_i$ and sub-Gaussian parameters $\sigma_i$.  Then
$$
\mathsf{P}\left\{{\sum_{i=1}^n (X_i-\mu_i) > t }\right\} < \exp\left({-\frac{t^2}{2\sum_{i=1}^n \sigma_i^2} }\right) \ .
$$
In particular, if the $X_i$ are bounded, $|X_i|<b$ then they are sub-Gaussian with parameter $b$ and the
inequality takes the form
$$
\mathsf{P}\left\{{\sum_{i=1}^n (X_i-\mu_i) > t }\right\} < \exp\left({-\frac{t^2}{2 n b^2} }\right) \ .
$$

==References==
* Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-108-49802-9

=Subgaussian distribution=
A probability distribution with tails comparable to those of the Gaussian distribution.  A random variable $X$ with mean $\mu$ is 
sub-Gaussian if there is a positive constant $\sigma$ such that
$$
\mathsf{E}[\exp(\lambda(X-\mu))] \le \exp(\lambda^2\sigma^2/2)
$$
for all real $\lambda$.

Clearly any normal random variable $N(\mu,\sigma^2)$ is sub-Gaussian with parameter $\sigma$.

A random variable is sub-Gaussian if and only if it has a moment generating function satisfying
$$
\mathsf{E}[\exp(\lambda X)] \le \exp(\lambda^2\sigma^2/2)
$$
for all real $\lambda$.

==References==
* Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-108-49802-9

=Markov inequality in probability theory=
A simple inequality giving a bound on the probability of deviation of a random variable with finite expectation from a given value.
Let $X(\omega)$ be a random variable.  Then 
$$
\mathsf{P}\{|X| > \epsilon\} \le \frac{\mathsf{E}X}{\epsilon} \ .
$$

The inequality follows from comparing the indicator function of the set $\{\omega : |X(\omega)| > \epsilon\}$ with the random variable $X/\epsilon$ and then taking expectations.

Application of Markov's inequality to the random variable $Y = (X-\mu)^2$ yields the [[Chebyshev inequality in probability theory]].

==References==
* Geoffrey Grimmett, David Stirzaker, "Probability and Random Processes" (4th ed), Oxford (2020) ISBN 0-19-884759-9
* Martin J. Wainwright, "High-dimensional Statistics", Cambridge (2019) ISBN 978-1-1-8-49802-9



=Steinitz exchange lemma=
A lemma on [[Linear independence|linearly independent]] sets and [[spanning set]]s in a vector space from which it is possible to deduce that dimension is well-defined.

Let $X = \{x_1,\ldots,x_m\}$ be a linearly independent set and $Y = \{y_1,\ldots,y_n\}$ a spanning set in a vector space $V$.   Then $m \le n$ and $Y$ can be re-ordered so that $\{x_1,\ldots,x_m,y_{m+1},\ldots,y_n$ is also a spanning set.

As a corollary, if $X$ and $Y$ are both bases (linearly independent and spanning) for $V$, then $n = m$.  Hence all bases for $V$ have the same size, and this shows that the dimension of $V$ is well-defined.

==References==
* P. M. Cohn, "Classic Algebra" Wiley (2000) ISBN 047187731X

=Spanning set=
''generating set'', ''for a module $M$ over a ring $R$''

A subset $S$ of $M$ such that every element of $M$ can be written as a finite linear combination $\sum_{i=1}^k r_i s_i$ with $r_i \in R$ and $s_i \in S$.

==References==
* P. M. Cohn, "Classic Algebra" Wiley (2000) ISBN 047187731X

=Edit distance=
A measure of dissimilarity between [[word]]s over some alphabet in terms of the number of elementary "edit" operations required to. turn one word into another.

Example include

* ''[[Hamming distance]]'' between words of the same length.  An edit operation consists of ''substitution'': replacing one letter in a given position by another letter in the same position.
* ''[[Lee distance]]'' between words of the same length over the alphabet $\mathbf{Z}/m$.  An edit operation consists of replacing one letter $i \pmod m$ by $i\pm 1 \pmod m$.

* ''Levenshtein distance'' between strings.  An edit operation consists of deleting one character or inserting one character.  A substitution can be obtained as a deletion following by an insertion, but may be considered as another elementary operation. 



==References==





=Star-semiring=
''$*$-semiring''

A [[semiring]] with a unary operator $*$.  

A '''Conway semiring''' satisfies the properties
$$
(x+y)^* = (x^*y)^*x^*
$$
and
$$
(xy)^* = 1 + x(yx)^*y \ .
$$

In a Conway semiring $C$ the $*$ operator extends to the matrix semiring over $C$.

==References==
Jean Berstel, Christophe Reutenauer, "Noncommutative rational series with applications", Encyclopedia of Mathematics and its Applications '''137''', Cambridge (2011) ISBN 978-0-521-19022-0 {{ZBL|1250.68007}}

=Club=
Let $\gamma$ be a [[limit ordinal]].  A subset $a \subset \gamma$ is unbounded in $\gamma$ if $\sup a = \gamma$; it is closed in $\gamma$ if it is closed in the [[order topology]] on $\gamma$: that is, for any limit $\mu < \gamma$ with $\sup(a\cap\mu) = \mu$ then $\mu \in a$.  A club ('''cl'''osed '''u'''n'''b'''ounded) subset $a$ is one which is both closed and unbounded in $\gamma$.

If the [[cofinality]] $\kappa$ of $\gamma$ is greater than $\omega$, then the intersection of any family of fewer than $\kappa$ clubs is again a club.

==References==
* Kenneth Kunen, "Set Theory", Studies in Logic '''34''', College Publications (2013) ISBN 978-1-84890-050-9. {{ZBL|1262.03001}}

=Weakly compact cardinal=
A cardinal $\mathfrak{k}$ is weakly compact if for any 2-colouring of the edges of a complete graph on a vertex set of cardinality $\mathfrak{k}$ there is a homogeneous subgraph of cardinality $\mathfrak{k}$.

Weakly compact cardinals are inaccessible: that is, their existence is independent of ZFC.

Every weakly compact cardinal is a [[Mahlo cardinal]].

==References==
* Erdős, Paul; Tarski, Alfred "On some problems involving inaccessible cardinals", Essays on the foundations of mathematics, Magnes Press (1961) pp. 50–82, {{MR|0167422}}. {{ZBL|0212.32502}}

=Weakly compact operator=
An operator $T : X \rightarrow Y$ on [[Banach space]]s is weakly compact if $T$ sends bounded subsets of  $X$
into weakly compact subsets of  $Y$.

See: [[Grothendieck space]], [[Dunford–Pettis property]], [[Dunford–Pettis operator]].

=Grothendieck universe=

A type of set in which all of mathematics can be performed.  Formally, a set $U$ with the properties:

*$x \in U$ and $y \in x$ implies $y \in U$: that is, $U$ is a [[transitive set]];
*If $x, y \in U$ then the [[doubleton]] $\{x,y\} \in U$;
*If $x \in U$ then the [[power set]] $\mathcal{P}(x) \in U$;
*If $\{x_\alpha\}_{\alpha \in y}$ is a family of elements of $U$ indexed by $y$, and $y \in U$, then the [[union]] $\cup_{\alpha \in y} x_\alpha \in U$.

The empty set is a Grothendieck universe, as is the set $V_\omega$ of [[hereditarily finite set]]s.

The existence of a non-trivial Grothendieck universe is equivalent to a [[large cardinal axiom]]: the existence of [[strongly inaccessible cardinal]]s.

== References ==
* Thomas Streicher, "Universes in Toposes" in Crosilla, Laura and Schuster, Peter (edd.) ''From sets and types to topology and analysis. Towards practicable foundations for constructive mathematics''. Clarendon Press (2005). pp. 78–90. ISBN 978-0-19-856651-9 {{ZBL|1092.03038}}

=Hereditarily finite set=
A finite set all of whose elements are finite sets.  

=Large cardinal axiom=

=Strongly inaccessible cardinal=

=Inaccessible cardinal=

=Finite type=

For an algebraic structure, an alternative term for ''finitely generated''.

A sheaf of modules $\def\cF{ {\mathcal F}}\cF$ over a sheaf of rings $\cO$ is a sheaf of finite type if it is locally generated over $\cO$ by a finite number of sections.

A Riemann surface $M$ is of finite type if it can be imbedded in a compact Riemann surface $\tilde M$ such that $\tilde M \setminus M$ consists of finitely many points. Cf. also [[Riemann surfaces, classification of|Riemann surfaces, classification of]] and (the references to) [[Double of a Riemann surface|Double of a Riemann surface]].


=Local properties of a group=
In [[group theory]], if $\mathcal{P}$ is a property of groups, then a group $G$ is said to be ''locally $\mathcal{P}$'' if every finitely-generated subgroup of $G$ has the property $\mathcal{P}$.  The term was introduced by A.G. Kurosh.

See
: [[Locally cyclic group]]
: [[Locally finite group]]
: [[Locally free group]]
: [[Locally nilpotent group]]
: [[Locally solvable group]]

The term "[[Locally normal group]]" does not fit this paradigm.

==References==
* B. Chandler, W. Magnus, "The History of Combinatorial Group Theory: A Case Study in the History of Ideas", Springer (2012) ISBN 1-4613-9487-2 
* A.G. Kurosh, Group theory. (Теория групп) (Russian) OGIZ, Moskva-Leningrad (1944) {{ZBL|0061.02101}}



=Separable partial order=
A partially ordered set $(X,{<})$ is '''Cantor separable''' if no strictly increasing linearly ordered subset has a least upper bound:
$$
a_1 < a_2 < \cdots < b
$$
implies there exists $c$ with
$$
a_1 < a_2 < \cdots < c < b \ .
$$
It is '''duBois–Reymond separable''' if a strictly increasing sequence can be separated from a decreasing sequence of upper bounds:
$$
a_1 < a_2 < \cdots < \cdots < b_2 < b_1
$$
implies there exists $c$ with
$$
a_1 < a_2 < \cdots < c < \cdots < b_2 < b_1 \ .
$$

==References==
* R.C. Walker, "The Stone–Čech compactification", Springer (1974)

=Heron triangle=
A triangle for which the lengths of the sides and the area are expressible by integers. Named after Heron (1st century A.D.), who studied triangles with side lengths $13,14,15$ and $5,12,13$, the areas of which are 84 and 30, respectively.

The Pythagorean triangles are special cases (cf. [[Pythagorean numbers|Pythagorean numbers]]).  In this case the area is a ''congruent number''.
 
The [[Heron formula]] for the area $S$ of a triangle in terms of its sides $a$, $b$ and $c$ and semi-perimeter $p=(a+b+c)/2$ is
$$S=\sqrt{p(p-a)(p-b)(p-c)},$$
so Heron triangles correspond to integer solutions to
$$
s^2 = p(p-a)(p-b)(p-c) \ .
$$

==References==


=Trace monoid=
Let $A$ be an [[alphabet]] with an irreflexive symmetric relation $I$ called ''independence''.  The complementary relation $I = A \times A \setminus I$ is the "dependence" relation.  Such an alphabet is a ''concurrence'' or ''dependency'' alphabet.  The free monoid on $A$ modulo the relations $ab=ba$ when $a,b \in I$ is the ''trace monoid'' on $(A,D)$.  The elements of a trace monoid are "traces" and the subsets are the "trace languages".

Trace monoids are used to model concurrency in computer languages.

==References==
* Diekert, Volker; Rozenberg, Grzegorz (edd) "The Book Of Traces" (World Scientific, 1995) ISBN  981-02-2058-8

=Trace-class operator=
An operator $T$ on a Hilbert space $H$ with complete orthonormal set $(e_n)$ for which the sum $\sum_n \langle Tx_n , x_n \rangle$ is finite.  For such operators, the ''trace'' is defined to be the value of this sum.  The set of trace-class operators on $H$ coincides with the set of squares of [[Hilbert-Schmidt operator]]s.  The trace-class operators are precisely the [[Schatten class]] for $p=1$.

==References==
* Retherford, J. R.  "Hilbert space: Compact operators and the trace theorem" London Mathematical Society Student Texts '''27'''. (Cambridge University Press, 1993) ISBN 0-521-42933-1. {{ZBL|0783.47031}}

=Schatten class=

''Schatten ideal''

A class of operators on a [[Hilbert space]].  Let $T$ be an operator with [[singular values]] $\sigma_n$.  For $1 \le p < \infty$ we say that $T$ is in the Schatten $p$-class if the sequence $(\sigma_n)$ is in $\ell_p$: that is, if $\sum_n |\sigma_n|^p$ converges, and then the $p$-root of the value is the Schatten $p$-norm of $T$.  The Schatten classes form ideals of the operator algebra.

The Schatten $2$-class is precisely the [[Hilbert–Schmidt operator]]s.  The Schatten $1$-class is the [[trace-class operator]]s.  

==References==
* Retherford, J. R.  "Hilbert space: Compact operators and the trace theorem" London Mathematical Society Student Texts '''27'''. (Cambridge University Press, 1993) ISBN 0-521-42933-1. {{ZBL|0783.47031}}
* Schatten, Robert. "Norm ideals of completely continuous operators" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, '''27''' . (Springer-Verlag, 1960)   {{ISBN|0090.09402}}

=Singular value=

The singular values of a complex matrix $A$ are the eigenvalues of $A^*A$, or equivalently of $AA^*$.  The '''singular value decomposition''' of $A$ is the expression $A=U\Sigma V$, with $U$ a unitary $(m\times n)$-matrix, $V$ a unitary $(n\times n)$-matrix and $\Sigma$ of the form
$$
\Sigma = \begin{pmatrix} {\mathcal D} & 0\\ 0 & 0\end{pmatrix},
$$
where ${\mathcal D}$ is diagonal with entries the singular values $s_1,\dots,s_k$ of $A$ and $k$ the rank of $A$.

In the case of a closed operator $A$ on a Hilbert space, then $A^*$ is a [[positive operator]] and the singular values of $A$ are the [[Spectrum of an operator|spectrum]] of $A^*A$.  

=Span=
'''Span''' may refer to 

* [[Linear hull]], also called ''linear span'' or ''span''
* [[Span (category theory)]]

=Span (category theory)=

A [[diagram]] in a [[category]] of the form
$$
\begin{array}{ccccc}
 & & C & & \\
 & f \swarrow & & \searrow g & \\
A & & & & B
\end{array}
$$

Two spans with arrows $(f,g)$ and $(f',g')$ are equivalent if for all $D,p,q$ the diagrams
$$
\begin{array}{ccccc}
 & & C & & \\
 & f \swarrow & & \searrow g & \\
A & & & & B \\
& p \searrow & & \swarrow q \\
& & D & & \\
\end{array}
\ \ \text{and}\ \ 
\begin{array}{ccccc}
 & & C & & \\
 & f' \swarrow & & \searrow g' & \\
A & & & & B \\
& p \searrow & & \swarrow q \\
& & D & & \\
\end{array}
$$
either both commute or both do not commute.

A [[pushout]] is the [[colimit]] of a span.

==References==
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971). ISBN 0-387-98403-8</TD></TR>
</table>

=Triple=
==Triple==
''monad, on a category  $  \mathfrak R $''

A [[Monoid|monoid]] in the [[Category|category]] of all endomorphism functors on  $  \mathfrak R $. 
In other words, a triple on a category  $  \mathfrak R $
is a covariant functor  $  T:  \mathfrak R \mathop \rightarrow \limits \mathfrak R $
endowed with natural transformations  $  \eta :  {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits T $
and  $  \mu :  T ^ {2} \mathop \rightarrow \limits T $ 
(here  $  {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $
denotes the identity functor on  $  \mathfrak R $) 
such that the following diagrams are commutative:

$$   \begin{array}{crclc} T (X)  & \mathop \rightarrow \limits ^ {T ( \eta _ {X} )}   &T ^ {2} (X)  & \mathop \leftarrow \limits ^ {\eta _ {T (X)}}   &T (X)  \\ {}  &{} _ {1 _ {T (X)}} \searrow  &\scriptsize {\mu _ {X}} \downarrow  &\swarrow _ {1 _ {T (X)}}  &{}  \\ {}  &{}  &T (X)  &{}  &{}  \\ \end{array}   $$

$$   \begin{array}{rcl} T ^ {3} (X)  & \mathop \rightarrow \limits ^ {T ( \mu _ {X} )}   &T ^ {2} (X)  \\ \scriptsize {\mu _ {T (X)}}  \downarrow  &{}  &\downarrow  \scriptsize {\mu _ {X}}  \\ T ^ {2} (X)  & \mathop \rightarrow \limits _ {\mu _ {X}}   &T (X)  \\ \end{array}   $$

A triple is sometimes called a standard construction, cf. [[#References|[2]]].

For any pair of adjoint functors  $  F :  \mathfrak R \mathop \rightarrow \limits \mathfrak L $
and  $  G:  \mathfrak L \mathop \rightarrow \limits \mathfrak R $ 
(see [[Adjoint functor|Adjoint functor]]) with unit and co-unit of adjunction  $  \eta :  {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits GF $
and  $  \epsilon :  FG \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $, 
respectively, the functor  $  T = GF:  \mathfrak R \mathop \rightarrow \limits \mathfrak R $
endowed with  $  \eta :  {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits T $
and  $  \mu = G ( \epsilon _ {F} ):  T ^ {2} \mathop \rightarrow \limits T $
is a triple on  $  \mathfrak R $. 
Conversely, for any triple  $  (T, \eta , \mu ) $
there exist pairs of adjoint functors  $  F $
and  $  G $
such that  $  T = GF $, 
and the transformations  $  \eta $
and  $  \mu $
are obtained from the unit and co-unit of the adjunction in the manner described above. The different such decompositions of a triple may form a proper class. In this class there is a smallest element (the Kleisli construction) and a largest element (the Eilenberg–Moore construction).

===Examples.===

1) In the category of sets, the functor which sends an arbitrary set to the set of all its subsets has the structure of a triple. Each set  $  X $
is naturally imbedded in the set of its subsets via singleton sets, and to each set of subsets of  $  X $
one associates the union of these subsets.

2) In the category of sets, every representable functor  $  H _ {A} (X) = H (A, X) $
carries a triple: The mapping  $  \eta _ {X} :  X \mathop \rightarrow \limits H (A, X) $
associates to each  $  x \in X $
the constant function  $  f _ {x} :  A \mathop \rightarrow \limits X $
with value  $  x $; 
the mapping  $  \mu _ {X} :  H (A, H (A, X)) \simeq H (A \times A, X) \mathop \rightarrow \limits H (A, X) $
associates to each function of two variables its restriction to the diagonal.

3) In the category of topological spaces, each topological group  $  G $, 
with unit  $  e $, 
enables one to define a functor  $  T _ {G} (X) = X \times G $
that carries a triple: Each element  $  x \in X $
is taken to the element  $  (x, e) $
and the mapping  $  \mu :  X \times G \times G \mathop \rightarrow \limits X \times G $
is defined by  $  \mu _ {X} (x, g, g ^  \prime  ) = (x, gg ^  \prime  ) $.

4) In the category of modules over a commutative ring  $  R $, 
each (associative, unital)  $  R $-
algebra  $  A $
gives rise to a triple structure on the functor  $  T _ {A} (X) = X \otimes _ {R} A $, 
in a manner similar to Example 3).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.F. Adams,   "Infinite loop spaces" , Princeton Univ. Press  (1978)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Godement,   "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.Sh. Tsalenko,   E.G. Shul'geifer,   "Categories"  ''J. Soviet Math.'' , '''7''' :  4  (1977)  pp. 532–586  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''13'''  (1975)  pp. 51–148</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.G. Manes,   "Algebraic theories" , Springer  (1976)</TD></TR></table>

====Comments====
The non-descriptive name  "triple"  for this concept has now largely been superseded by  "monad" , although there is an obstinate minority of category-theorists who continue to use it. A comonad (or cotriple) on a category  $  \mathfrak R $
is a monad on  $  \mathfrak R ^ {op} $; 
in other words, it is a functor  $  T:  \mathfrak R \mathop \rightarrow \limits \mathfrak R $
equipped with natural transformations  $  \epsilon :  T \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $
and  $  \delta :  T \mathop \rightarrow \limits T ^ {2} $
satisfying the duals of the commutative diagrams above. Every adjoint pair of functors ( $  F \dashv G $) 
gives rise to a comonad structure on the composite  $  FG $, 
as well as a monad structure on  $  GF $.

An important example of a functor which carries a comonad structure is  $  \Lambda :  {\mathop{\rm Ring}\nolimits} \mathop \rightarrow \limits {\mathop{\rm Ring}\nolimits} $, 
$  \Lambda (A)=1+tA[[t]] $, 
or, equivalently, the functor of big Witt vectors, cf. [[Lambda-ring| $  \lambda $-
ring]]; [[Witt vector|Witt vector]]. A special case of the natural transformation  $  W(A) \mathop \rightarrow \limits \Lambda (W(A)) $
occurs in algebraic number theory as the [[Artin–Hasse exponential]], [[#References|[a5]]].

Monads in the category of sets can be equivalently described by sets  $  T(n) $
of  $  n $-
ary operations for each cardinal number (or set)  $  n $; 
$  \eta _ {n} :  n \mathop \rightarrow \limits T(n) $
gives the projection operations  $  (x _ {1} , x _ {2} ,\dots) \mapsto x _ {i} $, 
and  $  \mu $
gives the rules for composing operations. See [[#References|[5]]] or [[#References|[a1]]]. This approach extends to monads in arbitrary categories, but it has not proved useful in general, as it has in or near sets.

Of the two canonical ways of constructing an adjunction from a given monad, mentioned in the main article above, the Eilenberg–Moore construction (or category of  $  T $-
algebras) is by far the more important. Given a monad  $  (T, \eta , \mu ) $
on a category  $  \mathfrak R $, 
a  $  T $-
algebra in  $  \mathfrak R $
is a pair  $  (A, \alpha ) $
where  $  \alpha :  TA \mathop \rightarrow \limits A $
is a morphism such that

$$   \begin{array}{lcr} A  \mathop \rightarrow \limits ^ {\eta _ {A}}   &TA  & \mathop \leftarrow \limits ^ {\mu _ {A}}  T ^ {2} A  \\ {} _ {1 _ {A}} \nwarrow  &\scriptsize \alpha  \downarrow  &\downarrow  \scriptsize {T _ {A}}  \\ {}  & A  & \mathop \leftarrow \limits _  \alpha   TA  \\ \end{array}   $$

commutes. A homomorphism of  $  T $-
algebras  $  (A, \alpha ) \mathop \rightarrow \limits (B, \beta ) $
is a morphism  $  f:  A \mathop \rightarrow \limits B $
in  $  \mathfrak R $
such that

$$   \begin{array}{rcl} TA  & \mathop \rightarrow \limits ^ {Tf}  &TB  \\ \scriptsize \alpha  \downarrow  &{}  &\downarrow  \scriptsize \beta  \\  A  &\mathop \rightarrow \limits _ {f}  & B  \\ \end{array}   $$

commutes; thus, one has a category  $  \mathfrak R ^ {T} $
of  $  T $-
algebras, with an evident forgetful functor  $  G ^ {T} :  \mathfrak R ^ {T} \mathop \rightarrow \limits \mathfrak R $. 
The functor  $  G ^ {T} $
has a left adjoint  $  F ^ { T} $, 
which sends an object  $  A $
of  $  \mathfrak R $
to the  $  T $-
algebra  $  (TA, \mu _ {A} ) $, 
and the monad induced by the adjunction ( $  F ^ { T} \dashv G ^ {T} $) 
is the one originally given.

Now the Kleisli category of  $  (T, \eta , \mu ) $
is just the full subcategory of  $  \mathfrak R ^ {T} $
on the objects  $  F ^ { T} (A) $: 
the category of free algebras (cf. also [[Category|Category]]).

For a monad  $  (T, \eta , \mu ) $
on  $  \mathfrak R $, 
in the Kleisli construction the category  $  \mathfrak L $
has as objects the objects of  $  \mathfrak R $, 
and as hom-sets the sets

$$  \mathfrak L (A, B)  =  \mathfrak R (A, TB).  $$

The composition rule for  $  \mathfrak L $
assigns to  $  f \in \mathfrak L (A, B) $
and  $  g \in \mathfrak L (B, C) $
the  $  \mathfrak R $-
composite:

$$  [A  \mathop \rightarrow \limits ^ {T}  TB  \mathop \rightarrow \limits ^ {T(g)}  TTC  \mathop \rightarrow \limits ^ {\mu _ {C}}  TC ]  \in  \mathfrak L (A, C);  $$

as identity mapping  $  1 _ {A} \in \mathfrak L (A, A) = \mathfrak R (T, TA) $
one uses the  $  \mathfrak R $-
morphism  $  \eta _ {A} :  A \mathop \rightarrow \limits TA $.

An adjoint pair  $  F:  \mathfrak R \mathop \rightarrow \limits \mathfrak L $, 
$  U:  \mathfrak L \mathop \rightarrow \limits \mathfrak R $
is obtained by setting  $  F(A)=A $
for  $  A \in \mathfrak R $,

$$  F(f)  =  \eta _ {B} \circ f :  A  \mathop \rightarrow \limits  B  \mathop \rightarrow \limits  TB  \in  \mathfrak R (A, TB)  =  \mathfrak L (A, B)  $$

for  $  f \in \mathfrak R (A, B) $, 
$  U(B)=TB $
for  $  B \in \mathfrak L $, 
and  $  U(g ) = \mu _ {G} \circ T(g) $
for  $  g \in \mathfrak L (B, C)= \mathfrak R (B, TC) $.

Then  $  \eta $
will serve as unit for the adjunction, while the co-unit  $  \epsilon :  FU \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak L} $
is given by

$$  \epsilon _ {B}  =  \mathop{\rm Id} _ {T(B)}  \in  \mathfrak R (TB, TB)  =  \mathfrak L (FUB, B).  $$

Co-algebras are defined in the same manner. In practice, co-algebras very often occur superposed on algebras; a comonad  $  G $
will be constructed on a category of algebras of some sort,  $  \mathfrak R $, 
leading to the category  $  {} ^ {G} \mathfrak R $
of bi-algebras. An important class of cases involves a monad  $  T $
and a cotriple  $  G $
on the same category  $  \mathfrak R $. 
There is a standard lifting of  $  G $
to a cotriple  $  G ^ {*} $
on  $  \mathfrak R ^ {T} $. 
A  "TG-bi-algebraTG-bi-algebra"  means an object of  $  {} ^ {G ^ {*}} ( \mathfrak R ^ {T} ) $; 
the reverse order is also possible, but rarely occurs, and the objects would not be called bi-algebras.

For the role of comonads in (algebraic) cohomology theories see [[Cohomology of algebras|Cohomology of algebras]] and [[#References|[a2]]], [[#References|[a3]]]; particularly [[#References|[a2]]] for explicit interpretation.

An adjunction is said to be monadic (or monadable) if the Eilenberg–Moore construction applied to the monad it induces yields an adjunction equivalent to the original one. Many important examples of adjunctions are monadic; for example, for any [[Variety of universal algebras|variety of universal algebras]], the forgetful functor from the variety to the category of sets and its left adjoint (the free algebra functor) form a monadic adjunction.

A monad  $  (T, \eta , \mu ) $
is said to be idempotent if  $  \mu $
is an isomorphism. In this case it can be shown that any  $  T $-
algebra structure  $  \alpha $
on an object  $  A $
is necessarily a two-sided inverse for  $  \eta _ {A} $, 
and hence that  $  \mathfrak R ^ {T} $
is isomorphic to the full subcategory  $  {\mathop{\rm Fix}\nolimits} (T) \subset  \mathfrak R $
consisting of all objects  $  A $
such that  $  \eta _ {A} $
is an isomorphism.  $  {\mathop{\rm Fix}\nolimits} (T) $
is a [[Reflective subcategory|reflective subcategory]] of  $  \mathfrak R $, 
the left adjoint to the inclusion being given by  $  T $
itself. Conversely, for any reflective subcategory of  $  \mathfrak R $, 
the monad on  $  \mathfrak R $
induced by the inclusion and its left adjoint is idempotent; thus, the adjunctions corresponding to reflective subcategories are always monadic.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,   C. Wells,   "Toposes, monads, and theories" , Springer  (1985)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W. Duskin,   "$K(\pi,n)$-torsors and the interpretation of  "monad"  cohomology"  ''Proc. Nat. Acad. Sci. USA'' , '''71'''  (1974)  pp. 2554–2557</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.W. Duskin,   "Simplicial methods and the interpretation of  "monad"  cohomology"  ''Mem. Amer. Math. Soc.'' , '''3'''  (1975)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Adamek,   H. Herrlich,   G.E. Strecker,   "Abstract and concrete categories" , Wiley (Interscience)  (1990)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Hazewinkel,   "Formal groups" , Acad. Press  (1978)  pp. Sects. 14.5; 14.6, E2</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Appelgate (ed.)  et al. (ed.) , ''Seminar on monads and categorical homology theory ETH 1966/7'' , ''Lect. notes in math.'' , '''80''' , Springer  (1969)</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Eilenberg,   J.C. Moore,   "Adjoint functors and monads"  ''Ill. J. Math.'' , '''9'''  (1965)  pp. 381–398</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  S. Eilenberg (ed.)  et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer  (1966)</TD></TR>
</table>

==Standard construction==
'''Standard construction''' is a concept in [[category theory]]. Other names are [[triple]], monad and functor-algebra.

Let $\mathfrak{S}$ be a [[category]]. A standard construction is a [[functor]] $T:\mathfrak{S} \rightarrow \mathfrak{S}$ equipped with natural transformations $\eta:1\rightarrow T$ and $\mu:T^2\rightarrow T$ such that the following diagrams commute:
$$
\begin{array}{ccc} 
T^3 Y & \stackrel{T\mu_Y}{\rightarrow} & T^2 Y \\
\mu_{TY}\downarrow& & \downarrow_Y \\
T^2 & \stackrel{T_y}{\rightarrow} & Y
\end{array}
$$
$$
\begin{array}{ccccc} 
TY & \stackrel{TY}{\rightarrow} & T^2Y & \stackrel{T_{\eta Y}}{\leftarrow} & TY \\
 & 1\searrow & \downarrow\mu Y & \swarrow1 & \\
 & & Y & & \\
\end{array}
$$

The basic use of standard constructions in topology is in the construction of various classifying spaces and their algebraic analogues, the so-called bar-constructions.

====References====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  J.M. Boardman,   R.M. Vogt,   "Homotopy invariant algebraic structures on topological spaces" , Springer  (1973)</TD></TR>
<TR><TD valign="top">[b2]</TD> <TD valign="top">  J.F. Adams,   "Infinite loop spaces" , Princeton Univ. Press  (1978)</TD></TR>
<TR><TD valign="top">[b3]</TD> <TD valign="top">  J.P. May,   "The geometry of iterated loop spaces" , ''Lect. notes in math.'' , '''271''' , Springer  (1972)</TD></TR>
<TR><TD valign="top">[b4]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)</TD></TR>
</table>



====Comments====
The term  "standard construction"  was introduced by R. Godement [[#References|[a1]]] for want of a better name for this concept. It is now entirely obsolete, having been generally superseded by  "monad"  (although a minority of authors still use the term  "triple" ). Monads have many other uses besides the one mentioned above, for example in the categorical approach to [[universal algebra]] (see [[#References|[a2]]], [[#References|[a3]]]).

====References====
<table>
<TR><TD valign="top">[c1]</TD> <TD valign="top">  R. Godement,   "Théorie des faisceaux" , Hermann  (1958)</TD></TR>
<TR><TD valign="top">[c2]</TD> <TD valign="top">  E.G. Manes,   "Algebraic theories" , Springer  (1976)</TD></TR>
<TR><TD valign="top">[c3]</TD> <TD valign="top">  M. Barr,   C. Wells,   "Toposes, triples and theories" , Springer  (1985)</TD></TR>
</table>

=Ostrowski representation=

MSC 11A67

Let $[a_1,a_2,\ldots]$ be the partial quotients of an infinite [[continued fraction]] and $(c_n)$ the corresponding [[continuant]]s, $c_0 = 1$, $c_1 = a_1$ and $c_{n+1} = a_{n+1} c_n + c_{n-1}$.  An Ostrowski representation of $N$ is
$$
N = \sum_{k=0}^n x_{k+1} c_k 
$$
where $0 \le x_k \le a_k$ and if $x_k = a_k$ then $x_{k-1} = 0$.  Every positive integer $M$ has a unique Ostrowski representation.

When the $a_n$ are all equal to $1$, the $c_n$ are the [[Fibonacci numbers]], and the Ostrowski representation is just the [[Zeckendorf representation]].  The addition of two $n$-digit numbers in Ostrowski representation based on a given continued fraction can be computed by three linear passes over the input.  

==References==
* Philipp Hieronymi; Alonza Terry jun.  "Ostrowski numeration systems, addition, and finite automata" ''Notre Dame J. Formal Logic'' '''59''' (2018) 215-232 {{ZBL|06870290}}

=Dirac comb=

A sum of [[Dirac delta-function]]s supported on a locally finite point set.  

==References==
* Michael Baake, Uwe Gromm; "Aperiodic order", vol.1, ''Encyclopedia of Mathematics and its Applications '''149''' (Cambridge, 2013) ISBN 978-0-521-86991-1  {{ZBL|1295.37001}}
* Marjorie Senechal; "Quasicrystals and geometry" (Cambridge, 1995) ISBN 0-521-57541-9  {{ZBL|0828.52007}}

=Delone set=

''Delaunay set'', $(r,R)$-''set''

A subset $D$ of $\mathbf{R}^n$ which is both discrete: there exists $r>0$ such that the balls of radius $r$ centred on points of $D$ are disjoint; and relatively dense: there exists $R$ such that the balls of radius $R$ centred on points of $D$ cover $\mathbf{R}^n$.

See also [[Covering and packing]].

The spectrum of a Delone set $D$ is defined as the [[Fourier transform]]
$$
\hat\gamma(s) = \lim_{T\to\infty} \frac{1}{(2T)^n} \sum_{d \in D_T} \exp(-2\pi i\, d\cdot s)
$$
where $D_T = D \cap [-T,T]^n$.  The spectrum $\hat\gamma$ is a positive measure and has a [[Lebesgue decomposition]] into a sum of discrete and continuous measures.  If
the discrete measure is supported on a countably infinite set $S$, then $D$ is said to satisfy the ''diffraction condition''.  

==References==
* Marjorie Senechal; "Quasicrystals and geometry" (Cambridge, 1995) ISBN 0-521-57541-9  {{ZBL|0828.52007}}

=SIS model=

A simple model in [[mathematical epidemiology]] which reduces to the logistic equation.  Assume that the population falls into two subgroups, "susceptible" ($S$) and "infected" ($I$), with susceptible members being infected at a rate proportional to the number of infected, and infected members recovering and returning to the susceptible subgroup at a constant rate.  We therefore have
$$
S' = - \beta S \cdot I + \alpha I
$$
and
$$
I' = \beta S \cdot I - \alpha I
$$
Since $S+I = N$ is constant, we have $I' = r I (1- k^{-1} I)$ where $r = \beta N - \alpha$ is the growth rate and $k = r / \beta$.  The basic reproduction number $R_0 = \beta N / \alpha$.  If $R_0 < 1$, so that $r<0$,  then $I$ decreases to zero.
Otherwise we have the explicit solution
$$
I(t) = \frac{ k B e^{rt} }{ 1 + B e^{rt} }
$$
where $B= I(0) / (k - I(0))$ and $I(t)$ tends to $k$ as $t \to \infty$

==Reference==
* Maia Martcheva, "An Introduction to Mathematical Epdidemiology" Texts in Applied Mathematics '''41''' (Springer, 2015) ISBN 978-1-4899-7611-6  {{ZBL|1333.92006}}

=Pregroup=
A pregroup generalises the notion of a free group with amalgamation.  A pregroup is a partially ordered monoid $(M,{\cdot},1,{\ge})$ with left and right adjoint maps $L$ and $R$ satisfying
$$
x^L \cdot x \ge 1 \ge x \cdot x^L
$$
$$
x \cdot x^R \ge 1 \ge x^R \cdot x 
$$
It follows that $x^{LR} = x = x^{RL}$

A [[partially ordered group]] is a pregroup, with both adjoint maps being group inversion.

==References==
* J. Stallings, "Group theory and three-dimensional manifolds" Yale Univ. Monogr. '''4''' (1971) {{ZBL|0241.57001}}

=Freiman homomorphism=

A map $\phi$ defined on a subset $A$ of an additive group $G$ to a group $H$ such that for $a_1,a_2,a_3,a-4 \in A$
$$
a_1 + a_2 = a_3 + a_4 \ \Rightarrow\ \phi(a_1) + \phi(a_2) = \phi(a_3) + \phi(a_4)
$$

Clearly an affine map $x \mapsto \psi(x) + b$ with $\psi$ a group homomorphism and $b \in H$ is a Frieman homomorphism.  A notable problem in [[additive combinatorics]] is to find conditions on $A$ that require every Freiman homomorphism to be affine.

More generally, a Freiman homomorphism of order $k$ satisfies the corresponding property for $k$-tuples with equal sums.

==References==
* Melvyn B. Nathanson, "Additive Number Theory: Inverse Problems and the Geometry of Sumsets", Graduate Texts in Mathematics '''165''' (Springer, 1996) ISBN 0-387-94655-1
* Terence Tao, Van H. Vu; "Additive Combinatorics", Cambridge Studies in Advanced Mathematics '''105''' (Cambridge University Press, 2006)ISBN 1-1394-5834-5
* David J. Grynkiewicz, "Structural Additive Theory", Developments in Mathematics '''30''' (Springer, 2013) ISBN 3-319-00416-6