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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
is a topological semigroup. Analytic semigroup – Type of strongly continuous semigroup Compact group – Topological group with compact topology Complete
Topological_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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
continuous group operations Topological module Topological ring Topological semigroup Topological vector space – Vector space with a notion of nearness Banaszczyk
Topological_abelian_group
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
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
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
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
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
R} is an additive topological group and a multiplicative topological semigroup. Topological rings are fundamentally related to topological fields and
Topological_ring
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
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
"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
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
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
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
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
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
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
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)
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
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
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
continuous group operations Topological module Topological ring Topological semigroup Topological vector space – Vector space with a notion of nearness
Complete_field
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
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
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
Mathematical model of the time dependence of a point in space
Because it is closed under composition, E(X, T) forms a compact right-topological sub-semigroup—meaning that for any fixed element, the right-multiplication
Dynamical_system
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)
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
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
semigroups. For example, profinite words are used to give an alternative characterization of the algebraic notion of a variety of finite semigroups.
Profinite_word
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
Property of measure-preserving dynamical systems
{\displaystyle \mathbb {R} _{+}} on compact metric spaces, with invariant measures taken with respect to the corresponding flow or semigroup. The transformation T {\displaystyle
Ergodicity
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
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
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
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
Probabilistic concept
to be a locally compact Polish space to guarantee the existence of regular conditional expectations, and denote its transition semigroup by ( P t ) t ≥
Doob's_h-transform
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
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
characterizes maximally dissipative operators as the generators of contraction semigroups. A dissipative operator has the following properties: From the inequality
Dissipative_operator
Swedish mathematician
isotone. Rådström characterized the generators of continuous semigroups of sets as compact convex sets. Rådström's Ph.D. students included Per Enflo and
Hans_Rådström
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
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)
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
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
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
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
American mathematics educator
University in 1974. She published her dissertation, Bisimple ω-semigroups in the locally compact setting, under the name Annie Laurer Alexander. It was supervised
Annie_Selden
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)
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
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
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
(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
Magma Module Monoid Monoid ring Quandle Quasigroup Quantum group Ring Semigroup Vector space Affine representation Character theory Great orthogonality
List_of_group_theory_topics
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)
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. Arithmetic operations
Binary_operation
subradius characterizes the minimal rate of growth of products in the semigroup generated by M {\displaystyle {\mathcal {M}}} . The p-radius characterizes
Joint_spectral_radius
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
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
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
Sometimes, especially when defining completely monotonic functions on semigroups, they are defined as functions f {\displaystyle f} such that ∇ a 1 … ∇
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
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
Bimodal function
kernel Christian Berg, Christensen, Paul Ressel. Harmonic Analysis on Semigroups, GTM, Springer Verlag. Z. Sasvári, Positive Definite and Definitizable
Positive-definite_function
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
Open conjecture in multiplicative Diophantine approximation
, β {\displaystyle x_{\alpha ,\beta }} under an appropriate positive semigroup in the diagonal group. This connection permits the use of ergodic theory
Littlewood_conjecture
Czech mathematician (1933–2018)
PhD from Prague's Charles University in 1963. His thesis on commutative semigroups was supervised by Miroslav Katětov. Hedrlín held the title of Docent (associated
Zdeněk_Hedrlín
Polish mathematician (1930–2005)
variables in compact groups, connections between measurability and connectivity, generalized convolutions, and decomposability semigroups. He also studied
Kazimierz_Urbanik
Type of random process
sufficient condition for a QSD to exist is given considering the transition semigroup ( P t , t ≥ 0 ) {\displaystyle (P_{t},t\geq 0)} of the process before
Quasi-stationary_distribution
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
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
Mathematical objects
locally compact groups and manifolds. Let X {\displaystyle X} be an arbitrary nonempty set and let G {\displaystyle G} be a commutative semigroup acting
Boehmians
Chinese-American mathematician (born 1949)
Zbl 1079.60005. Wang, Feng-Yu (2005). Functional inequalities, Markov semigroups and spectral theory. Beijing/New York: Science Press. doi:10.1016/B978-0-08-044942-5
Shing-Tung_Yau
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
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
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
Partial differential equation describing the evolution of temperature in a region
dissipative, thus by the spectral theorem it generates a one-parameter semigroup. In the special cases of propagation of heat in an isotropic and homogeneous
Heat_equation
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
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
Mathematical function often applied to matrices
of the logarithmic norms may be infinite. Further variants of use in semigroup theory are based on the limit M [ A ] = lim h → 0 + ‖ e h A ‖ − 1 h ,
Logarithmic_norm
American mathematician
assumption of dynamics is that one has a phase space and some group or semigroup of self-maps of that space that play the role of describing time evolution
Daniel_Rudolph
travel, tourism, insurance
COMPACT SEMIGROUP
COMPACT SEMIGROUP
Boy/Male
Hindu, Indian, Sanskrit
In the Company
Girl/Female
Arabic, Muslim
Beauty of Company
Boy/Male
Indian, Punjabi, Sikh
Liberation through Company
Boy/Male
Indian, Tamil
No Compare
Girl/Female
Arabic, Muslim
Beauty of Company
Girl/Female
Indian, Telugu
Good Company
Boy/Male
Indian, Punjabi, Sikh
Company of Guru
Boy/Male
Hindu, Indian
Compact; Safe; Secure
Girl/Female
Arabic
Sensible Contact
Boy/Male
Indian, Punjabi, Sikh
Lord's Company
Girl/Female
Hindu, Indian
Compare
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
Hindu, Indian, Sanskrit
Company
Girl/Female
Hindu, Indian, Marathi, Tamil
Compact; Promise
Boy/Male
Indian, Punjabi, Sikh
Good Company
Boy/Male
Hindu, Indian
Compact; Firm; Solid
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
Muslim
Beauty of company
Girl/Female
Tamil
Compare
Boy/Male
Indian, Sanskrit
Fallen from Glory
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
COMPACT SEMIGROUP
travel, tourism, insurance