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mathematics, a compact semigroup is a semigroup in which the sets of solutions to equations can be described by finite sets of equations. The term "compact" here
Compact_semigroup
Generalization of the exponential function
In mathematical analysis, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function
C0-semigroup
Algebraic structure
appears in the theory of one-parameter operator semigroups: see C0-semigroup. The binary operation of a semigroup is most often denoted multiplicatively: x
Semigroup
Compact topological semigroup
Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such that S is a compact space and the product is semi-continuous, then S has
Ellis–Numakura_lemma
Type of topological space in mathematics
space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it
Locally_compact_space
is a topological semigroup. Analytic semigroup – Type of strongly continuous semigroup Compact group – Topological group with compact topology Complete
Topological_semigroup
Generalized function whose value is zero everywhere except at zero
and compactly supported, but not a mollifier because it is not smooth. Approximations to the delta functions often arise as convolution semigroups. This
Dirac_delta_function
Branch of mathematics
twentieth century. The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid G satisfying the following properties:
Abstract analytic number theory
Abstract_analytic_number_theory
Algebraic structure with an associative operation and an identity element
with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of
Monoid
holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the
Loewner_differential_equation
Special types of subgroups encountered in group theory
apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation)
Centralizer_and_normalizer
Differential operator in mathematics
is a strongly continuous contraction semigroup whose generator is the Laplacian; more generally, the heat semigroup acts contractively on Lp for 1 ≤ p ≤
Laplace_operator
Type of topological group in mathematics
mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important
Locally_compact_group
Stochastic process
sup norm is a Banach space. A Feller semigroup on C 0 ( X ) {\textstyle C_{0}(X)} is a contraction C0-semigroup of positive operators on C 0 ( X ) {\textstyle
Feller_process
In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such
Paratopological_group
series. The semigroup is made up of those elements in the complexification which, when acting on the Hermitian symmetric space of compact type, leave
Invariant_convex_cone
Group of 𝑛 × 𝑛 invertible matrices
monoid, but occasionally also full linear semigroup, general linear monoid etc. It is actually a regular semigroup. The infinite general linear group or stable
General_linear_group
Representation theory of the symplectic group
representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been
Oscillator_representation
Property of topological spaces
space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below
Compactly_generated_space
Branch of mathematics that studies algebraic structures
lemma Semigroup Subsemigroup Free semigroup Green's relations Inverse semigroup (or inversion semigroup, cf. [1]) Krohn–Rhodes theory Semigroup algebra
List of abstract algebra topics
List_of_abstract_algebra_topics
subset of E {\displaystyle E} that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of K {\displaystyle
Ryll-Nardzewski fixed-point theorem
Ryll-Nardzewski_fixed-point_theorem
Functional equation characterizing associative binary operations
associative in the usual algebraic sense, and therefore underlies the study of semigroups and many kinds of aggregation operators. When additional regularity conditions
Associativity_equation
Concept in topology
\beta S} . This turns β S {\displaystyle \beta S} into a compact right topological semigroup. The algebraic structure of β S {\displaystyle \beta S} —specifically
Stone–Čech_compactification
"The category of CGWH spaces" (PDF). Lawson, J; Madison, B (1974). "Quotients of k-semigroups". Semigroup Forum. 9: 1–18. doi:10.1007/BF02194829. v t e
Weak_Hausdorff_space
Group that is a topological space with continuous group operations
descriptions of redirect targets Topological module Topological ring Topological semigroup Topological vector space – Vector space with a notion of nearness i.e
Topological_group
Memoryless property of a stochastic process
collection ( P t ) t ≥ 0 {\displaystyle (P_{t})_{t\geq 0}} its transition semigroup. There exists multiple alternative formulations of the elementary Markov
Markov_property
R} is an additive topological group and a multiplicative topological semigroup. Topological rings are fundamentally related to topological fields and
Topological_ring
Mathematical category formed by reversing morphisms
Given a semigroup (S, ·), one usually defines the opposite semigroup as (S, ·)op = (S, *) where x*y ≔ y·x for all x,y in S. So also for semigroups there
Opposite_category
Algebraic ring without a multiplicative identity
and multiplication such that (R, +) is an abelian group, (R, ·) is a semigroup, Multiplication distributes over addition. A rng homomorphism is a function
Rng_(algebra)
continuous group operations Topological module Topological ring Topological semigroup Topological vector space – Vector space with a notion of nearness Banaszczyk
Topological_abelian_group
Mathematical inequality about the convolution of two functions
that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the L 2 {\displaystyle L^{2}} norm (that is, the Weierstrass
Young's convolution inequality
Young's_convolution_inequality
One of two theorems in dynamical systems
t\geq 0,} be the transition probabilities for a time-homogeneous Markov semigroup on X, i.e. Pr [ X t ∈ A | X 0 = x ] = P t ( x , A ) . {\displaystyle \Pr[X_{t}\in
Krylov–Bogolyubov_theorem
Nonlocal mathematical operator
B_{r}(x)}{{\frac {f(x)-f(y)}{|x-y|^{d+2s}}}\,dy}} Using the fractional heat-semigroup which is the family of operators { P t } t ∈ [ 0 , ∞ ) {\displaystyle
Fractional_Laplacian
Proof that every structure with certain properties is isomorphic to another structure
of copies of A. In the study of semigroups, the Wagner–Preston theorem provides a representation of an inverse semigroup S, as a homomorphic image of the
Representation_theorem
Field of mathematics
a continuous transformation, a continuous flow, or more generally, a semigroup of continuous transformations of that space. The origins of topological
Topological_dynamics
words, an idempotent measure is an idempotent element in the topological semigroup of probability measures on the given metric group. Explicitly, given a
Idempotent_measure
Fundamental solution to the heat equation, given boundary values
spectral mapping theorem gives a representation of T in the form the semigroup T = e t Δ . {\displaystyle T=e^{t\Delta }.} There are several geometric
Heat_kernel
continuous group operations Topological module Topological ring Topological semigroup Topological vector space – Vector space with a notion of nearness
Complete_field
Specific element of an algebraic structure
or "unity." In the example S = {e,f} with the equalities given, S is a semigroup. It demonstrates the possibility for (S, ∗) to have several left identities
Identity_element
Algebra describing information processing
, D ) {\displaystyle (\Phi ,D)} : Where Φ {\displaystyle \Phi } is a semigroup, representing combination or aggregation of information, and D {\displaystyle
Information_algebra
Type of vector space in math
states the following: If Ut is a (strongly continuous) one-parameter semigroup of unitary operators on a Hilbert space H, and P is the orthogonal projection
Hilbert_space
Lie group homomorphism from the real numbers
real line. Exponential map (Lie theory) Integral curve One-parameter semigroup Noether's theorem The Wikibook Abstract Algebra has a page on the topic
One-parameter_group
Branch of mathematics that studies dynamical systems
result also extends to the case of strongly continuous one-parameter semigroup of contractive operators on a reflexive space. Remark: Some intuition
Ergodic_theory
Motion of particles in a fluid
boundary condition. The mathematical setting for this problem can be the semigroup approach. To use this tool, we introduce the unbounded operator ΔD defined
Flow_(mathematics)
Potential in mathematics
|^{-\alpha }{\hat {f}}(\xi ).} The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions I α
Riesz_potential
Theorem in probability theory
{\displaystyle X} be a locally compact Abelian group. Denote by M 1 ( X ) {\displaystyle M^{1}(X)} the convolution semigroup of probability distributions
Raikov's_theorem
Type of differential operator
eigenvectors of L. (See Spectral theorem.) Generates a semigroup on L2(U): −L generates a semigroup { S ( t ) ; t ≥ 0 } {\displaystyle \{S(t);t\geq 0\}}
Elliptic_operator
Smallest convex set containing a given set
Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The
Convex_hull
Theorem relating unitary operators to one-parameter Lie groups
theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces. Hall 2013 Theorem 10.15 Hall, B.C. (2013)
Stone's theorem on one-parameter unitary groups
Stone's_theorem_on_one-parameter_unitary_groups
Mathematical model of the time dependence of a point in space
possible to model time evolution: T ^ {\displaystyle {\hat {T}}} can be a semigroup with one parameter t {\displaystyle t} called time that will also belong
Dynamical_system
Partial order with joins
speak simply of semilattices. A semilattice is a commutative, idempotent semigroup; i.e., a commutative band. A bounded semilattice is an idempotent commutative
Semilattice
Locally compact topological group with an invariant averaging operation
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant
Amenable_group
Random process independent of past history
X} and ( P t ) t ≥ 0 {\displaystyle (P_{t})_{t\geq 0}} the transition semigroup of the process. Transition functions are generalizations of the transition
Markov_chain
Type of topological space
∈ X if x lies in the interior of some compact subset of X. X is a locally compact space if it is locally compact at every point in the space. A proper
Polyadic_space
Concept in mathematics regarding sets operating on groups
analogous to that of compactness in topology, and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set"
Group_with_operators
Nonempty, upper-bounded, downward-closed subset
Non-empty family of sets that is closed under finite unions and subsets Semigroup ideal Boolean prime ideal theorem – Ideals in a Boolean algebra can be
Ideal_(order_theory)
semigroups. For example, profinite words are used to give an alternative characterization of the algebraic notion of a variety of finite semigroups.
Profinite_word
Branch of mathematics
specialized structure by adding constraints. For example, a magma becomes a semigroup if its operation is associative. Homomorphisms are tools to examine structural
Algebra
Algebraic structure with addition, multiplication, and division
F(X) is very close to X: if X is smooth and proper (the analogue of being compact), X can be reconstructed, up to isomorphism, from its field of functions
Field_(mathematics)
Transformations induced by a mathematical group
does not define bijective maps and equivalence relations however. See semigroup action. Instead of actions on sets, we can define actions of groups and
Group_action
Vector space with a notion of nearness
continuous group operations Topological module Topological ring Topological semigroup Topological vector lattice Measure theory in topological vector spaces –
Topological_vector_space
Magma Module Monoid Monoid ring Quandle Quasigroup Quantum group Ring Semigroup Vector space Affine representation Character theory Great orthogonality
List_of_group_theory_topics
Swedish logician, philosopher, and mathematical statistician
theorem on a locally compact group. Teor. Verojatnost. i Primenen. 10 1965 367–371. Martin-Löf, Per Probability theory on discrete semigroups. Z. Wahrscheinlichkeitstheorie
Per_Martin-Löf
Group in group theory and physics
{\mathcal {L}}=-\sum _{j=1}^{n}(X_{j}^{2}+Y_{j}^{2}),} the corresponding heat semigroup is generated by − 1 2 L {\displaystyle -{\frac {1}{2}}{\mathcal {L}}}
Heisenberg_group
Construction in category theory
construction may be carried out if the A i {\displaystyle A_{i}} 's are sets, semigroups, topological spaces, rings, modules (over a fixed ring), algebras (over
Inverse_limit
Relation in theoretical computer science
their unknowns are erased; as such, they are usually studied over free semigroups. quadratic equations, which are those containing each of their unknowns
Word_equation
Dutch mathematician and professor (1936–2020)
de Groot as her supervisor. She defended her PhD thesis "Topological Semigroups" and obtained her degree in 1960, also cum laude. In 1980 she became a
Ietje_Paalman-de_Miranda
Abelian group extending a commutative monoid
of M". This is known as the "group completion of a semigroup" or "group of fractions of a semigroup". In the language of category theory, any universal
Grothendieck_group
Stochastic differential equation
Feller process ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} with Feller semigroup T = ( T t ) t ≥ 0 {\displaystyle T=(T_{t})_{t\geq 0}} and state space
Infinitesimal generator (stochastic processes)
Infinitesimal_generator_(stochastic_processes)
Irish mathematician (1948–2006)
(1997), 367–374. Averaging theorems for linear operators in compact groups and semigroups, Studia Math., 124 (1997), 249—258 (with T.T. West). Products
Gerard_Murphy_(mathematician)
One-to-one correspondence
(1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4. John Meakin (2007). "Groups and semigroups: connections
Bijection
Vector space of functions in mathematics
ISBN 978-3-642-15563-5, MR 2777530, Zbl 1217.46002. Lunardi, Alessandra (1995), Analytic semigroups and optimal regularity in parabolic problems, Basel: Birkhäuser Verlag
Sobolev_space
characterizes maximally dissipative operators as the generators of contraction semigroups. A dissipative operator has the following properties: From the inequality
Dissipative_operator
Theorem in category theory
show that the topos T has finite colimits. The forgetful functor from semigroups to sets is monadic. This functor does not preserve arbitrary coequalizers
Beck's_monadicity_theorem
American mathematician (born 1941)
with Donald E. Ramirez: Representations of commutative semitopological semigroups. Springer-Verlag. 1975. ISBN 039027819X. Dunkl, Charles F. (1984). "Orthogonal
Charles_F._Dunkl
Theorem about projections of coadjoint orbits of a connected compact Lie group
theorem states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a
Kostant's_convexity_theorem
Important problem in lattice theory
Grillet, Pierre Antoine (1976). "Directed colimits of free commutative semigroups". Journal of Pure and Applied Algebra. 9 (1): 73–87. doi:10.1016/0022-4049(76)90007-4
Congruence_lattice_problem
Algebraic structure
elements of a semifield form a group. However, the pair (S,+) is only a semigroup, i.e. additive inverse need not exist, or, colloquially, 'there is no
Semifield
Swiss mathematician and physicist (1939–2015)
ed. (2006). "Laplacians on metric graphs: eigenvalues, resolvents and semigroups by Vadim Kostrykin and Robert Schrader". Quantum Graphs and Their Applications:
Robert_Schrader
Canadian-American mathematician of Greek origin and operations researcher (1914–1981)
g x T g {\displaystyle \sum _{g}\lambda _{g}xT_{g}} for some group or semigroup G of linear operators T g {\displaystyle T_{g}} on a Banach space E converge)
Leonidas_Alaoglu
American mathematician (born 1931)
114685, 37. Charalambous, Nelia; Gross, Leonard: The Yang-Mills heat semigroup on three-manifolds with boundary. Comm. Math. Phys. 317 (2013), no. 3
Leonard_Gross
Real numbers with + and - infinity added
defined above, R ¯ {\displaystyle {\overline {\mathbb {R} }}} is not even a semigroup, let alone a group, a ring or a field as in the case of R {\displaystyle
Extended_real_number_line
Lions–Lax–Milgram theorem (partial differential equations) Lumer–Phillips theorem (semigroup theory) Marcinkiewicz theorem (functional analysis) Mazur–Ulam theorem
List_of_theorems
Mathematical operation with two operands
keystone of most structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces. More precisely, a
Binary_operation
Lie group of complex numbers of unit modulus; topologically a circle
then the orbit is dense, and in fact equidistributed. Similarly, the semigroup of translations R a , R a 2 , … {\displaystyle R_{a},R_{a}^{2},\dots }
Circle_group
(Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1965 and updated every 2 to 4 years since.
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Concept in mathematical theory of categories
category of groups is a reflective subcategory of the category of inverse semigroups. Similarly, the category of commutative associative algebras is a reflective
Reflective_subcategory
Mathematical concept
contraction semigroup on L2(R2). Since Py is positive and integrable with integral 1, the operators Ts also define a contraction semigroup on each Lp space
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
Finite or infinite ordered list of elements
more elements of A, with the binary operation of concatenation. The free semigroup A+ is the subsemigroup of A* containing all elements except the empty
Sequence
Compact interpolation between Banach spaces. New Brunswick, NJ: Rutgers University. Gill, Tepper L (1974). Tensor products of contraction semigroups on
List of African-American mathematicians
List_of_African-American_mathematicians
Complex Analysis, Fixed-points and Iterations of Holomorphic Mappings
Mathematics, Springer-Verlag, ISBN 0-387-94067-7 Shoikhet, D. (2001), Semigroups in geometrical function theory, Kluwer Academic Publishers, ISBN 0-7923-7111-9
Denjoy–Wolff_theorem
Branch of mathematics
often studied using one-parameter families of operators, such as operator semigroups, which generalize the exponential function from numbers or matrices to
Mathematical_analysis
Mathematical theorem
Stone's theorem on one-parameter unitary groups Hille–Yosida theorem C0-semigroup [xn, p] = i ℏ nxn − 1, hence 2‖p‖ ‖x‖n ≥ n ℏ ‖x‖n − 1, so that, ∀n: 2‖p‖ ‖x‖
Stone–von_Neumann_theorem
partitions of 12 white objects and 3 black ones 1915 = number of nonisomorphic semigroups of order 5 1916 = sum of first 50 composite numbers 1917 = number of partitions
1000_(number)
Characterization of how many integers are prime
prime number theorem and disjointness of additive and multiplicative semigroup actions. Duke Mathematical Journal, 171(15), 3133-3200. Avigad, Jeremy;
Prime_number_theorem
Australian mathematician
George A. (2020). "A graph-theoretic description of scale-multiplicative semigroups of automorphisms". Israel Journal of Mathematics. 237: 221–265. arXiv:1710
George_A._Willis
course titles. Abstract analytic number theory The study of arithmetic semigroups as a means to extend notions from classical analytic number theory. Abstract
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
from a collection of subsets of a set X {\displaystyle X} to an abelian semigroup. For example, Lebesgue measure is a valuation on finite unions of convex
Valuation_(geometry)
Function from sets to numbers
of sets is modular. In geometry, a set function valued in some abelian semigroup that possess this property is known as a valuation. This geometric definition
Set_function
(E,{\mathcal {E}}^{*})} , which uniquely associated with the Markovian semigroup { P t : t ∈ [ 0 , ∞ ) } {\displaystyle \{P_{t}:t\in [0,\infty )\}} . Consequently
Borel_right_process
COMPACT SEMIGROUP
COMPACT SEMIGROUP
Girl/Female
Arabic
Sensible Contact
Surname or Lastname
Americanized spelling of German Kahle. Compare Kahley or Köhler (see Kohler).English and Manx
Americanized spelling of German Kahle. Compare Kahley or Köhler (see Kohler).English and Manx : variant spelling of Caley.
Boy/Male
Indian, Punjabi, Sikh
Good Company
Girl/Female
Hindu, Indian
Compare
Boy/Male
Hindu, Indian, Sanskrit
Company
Boy/Male
Hindu, Indian
Compact; Safe; Secure
Boy/Male
Indian, Punjabi, Sikh
Lord's Company
Surname or Lastname
Americanized form of German Eisele. Compare Isley.English
Americanized form of German Eisele. Compare Isley.English : unexplained. This name is quite widespread in Britain.
Girl/Female
Arabic, Muslim
Beauty of Company
Girl/Female
Indian, Telugu
Good Company
Boy/Male
Indian, Punjabi, Sikh
Liberation through Company
Girl/Female
Hindu, Indian, Marathi, Tamil
Compact; Promise
Girl/Female
Arabic, Muslim
Beauty of Company
Boy/Male
Hindu, Indian, Sanskrit
In the Company
Boy/Male
Indian, Punjabi, Sikh
Company of Guru
Boy/Male
Indian, Tamil
No Compare
Boy/Male
Indian, Sanskrit
Fallen from Glory
Boy/Male
Hindu, Indian
Compact; Firm; Solid
Girl/Female
Muslim
Beauty of company
Girl/Female
Tamil
Compare
COMPACT SEMIGROUP
COMPACT SEMIGROUP
Girl/Female
Biblical
That is poor.
Girl/Female
Indian
Fire horse, Grace
Girl/Female
English
Caprice.
Boy/Male
Arabic, Muslim
A Person who Upholds the Truth; Just
Male
Italian
Italian name of Germanic origin, derived from the element helm, ELMO means "helmet, protection."Â
Boy/Male
Indian, Punjabi, Sikh
Victorious Lamp of God
Boy/Male
Indian, Telugu
Handsome; Precious; Graceful
Boy/Male
Arabic, Muslim
Struggle; Fight
Boy/Male
Arthurian Legend
Name of a king.
Boy/Male
Australian, Christian, Gaelic, Irish
Rock; Comely; Little Rock; Handsome
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
v. i.
To be like or equal; to admit, or be worthy of, comparison; as, his later work does not compare with his earlier.
a.
Compact; pressed close; concentrated; firmly united.
n.
Extent; reach; sweep; capacity; sphere; as, the compass of his eye; the compass of imagination.
imp. & p. p.
of Compact
n.
An inclosing limit; boundary; circumference; as, within the compass of an encircling wall.
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.
p. pr. & vb. n.
of Compact
a.
Strong; firm; compact.
v. t.
To mingle, as different fertilizing substances, in a mass where they will decompose and form into a compost.
n.
A mixture for fertilizing land; esp., a composition of various substances (as muck, mold, lime, and stable manure) thoroughly mingled and decomposed, as in a compost heap.
adv.
In a compact manner; with close union of parts; densely; tersely.
n.
Contact or impression by touch; collision; forcible contact; force communicated.
v. t.
To compact or join anew.
n.
An association of persons for the purpose of carrying on some enterprise or business; a corporation; a firm; as, the East India Company; an insurance company; a joint-stock company.
n.
One who makes a compact.
v. i.
To bear or endure; to put up (with); as, to comport with an injury.
n.
The crew of a ship, including the officers; as, a whole ship's company.
v. t.
To manure with compost.
n.
Guests or visitors, in distinction from the members of a family; as, to invite company to dine.