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Topological continuum undefinable as the union of any two proper subcontinua
definition of indecomposability. In 1922 Bronisław Knaster described the pseudo-arc, the first example found of a hereditarily indecomposable continuum.
Indecomposable_continuum
Topics referred to by the same term
Look up indecomposability or indecomposable in Wiktionary, the free dictionary. Indecomposability or indecomposable may refer to any of several subjects
Indecomposability
In intuitionistic analysis and in computable analysis, indecomposability or indivisibility (German: Unzerlegbarkeit, from the adjective unzerlegbar) is
Indecomposability (intuitionistic logic)
Indecomposability_(intuitionistic_logic)
algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Indecomposable is a weaker notion
Indecomposable_module
Probability distribution
_{n=1}^{\infty }2^{n}D_{n},} and each term in this sum is indecomposable. At the other extreme from indecomposability is infinite divisibility. Cramér's theorem shows
Indecomposable_distribution
Statement in abstract algebra
) {\displaystyle R/(q_{i})} are indecomposable, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any β , γ < α {\displaystyle
Additively indecomposable ordinal
Additively_indecomposable_ordinal
Concept in point-set topology
union of all proper subcontinua of A that contain p. If a continuum is indecomposable, then its composants are pairwise disjoint. The composants of a continuum
Composant
Type of group and algebra representation
irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations)
Irreducible_representation
Axiomatic set theories based on the principles of mathematical constructivism
properties. This is well studied in intuitionistic analysis. The so-called indecomposability schema U Z {\displaystyle {\mathrm {UZ} }} (Unzerlegbarkeit) for set
Constructive_set_theory
Branch of mathematics that studies abstract algebraic structures
subrepresentations, it is said to be decomposable. Otherwise, it is said to be indecomposable. In favorable circumstances, every finite-dimensional representation
Representation_theory
Causal relationships between points in a manifold
point in M {\displaystyle M} is called a terminal indecomposable past set (TIP). A proper indecomposable past set (PIP) is an IP which isn't a TIP. I − (
Causal_structure
Mathematical object in abstract algebra
Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module
Injective_module
Generalization of the Cartesian product
In mathematics, the direct product of a collection of algebraic structures (such as groups, rings, or vector spaces) is a structure of the same type constructed
Direct_product
Nonempty compact connected metric space
proper subcontinua. A continuum X is hereditarily indecomposable if every subcontinuum of X is indecomposable. The dimension of a continuum usually means its
Continuum_(topology)
Dutch mathematician and logician
approximation theorem Bar induction Degree of a continuous mapping Indecomposability Indecomposable continuum Invariance of domain Spread Proving hairy ball theorem
L._E._J._Brouwer
Infinite ordinal number class
upgraded to a higher level). Additively indecomposable A limit ordinal α is called additively indecomposable if it cannot be expressed as the sum of β
Limit_ordinal
Concept in functional analysis
{\displaystyle X,} indecomposable Banach spaces are prime. The most well-known example of indecomposable spaces are in fact hereditarily indecomposable, which means
Complemented_subspace
Studies linear representations of finite groups over fields of positive characteristic
indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules: the socle of each projective indecomposable
Modular_representation_theory
Number divisible only by 1 and itself
branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field of a given
Prime_number
in the area of abstract algebra known as module theory, a principal indecomposable module has many important relations to the study of a ring's modules
Principal indecomposable module
Principal_indecomposable_module
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed
List_of_prime_knots
Factorization under function composition
decomposable in this way are composite polynomials; those which are not are indecomposable polynomials or sometimes prime polynomials (not to be confused with
Polynomial_decomposition
Abstract algebra concept
semisimple if and only if every module over it is a semisimple module. An indecomposable module is a module that is not a direct sum of two nonzero submodules
Decomposition_of_a_module
Mathematical theorem
of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. We say that a group G satisfies the ascending chain condition
Krull–Schmidt_theorem
Three disjoint sets that share a common boundary
Brouwer (1910) of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum. The Lakes
Lakes_of_Wada
Function that preserves distinctness
set in the reals cannot have a left inverse, as it would violate indecomposability, by giving a retraction of the real line to the set {0,1}. Williams
Injective_function
Means of constructing a group from two subgroups
a group cannot be expressed as such a direct sum then it is called indecomposable. A group G is called the direct sum of two subgroups H1 and H2 if each
Direct_sum_of_groups
Algebraic theory
split C is indecomposable and any homomorphism from an indecomposable module to C that is not an isomorphism factors through B. A is indecomposable and any
Auslander–Reiten_theory
Matrices named after Élie Cartan
and j ∉ I {\displaystyle j\notin I} . A is indecomposable if it is not decomposable. Let A be an indecomposable generalized Cartan matrix. We say that A
Cartan_matrix
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the
Prime_knot
Type of topological continuum
topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and
Pseudo-arc
Generalization of vector spaces from fields to rings
Historically these modules are also called completely reducible. Indecomposable An indecomposable module is a non-zero module that cannot be written as a direct
Module_(mathematics)
Algebraic geometry term
In algebraic geometry, Horrocks bundles are certain indecomposable rank 3 vector bundles (locally free sheaves) on 5-dimensional projective space, found
Horrocks_bundle
Operations on ordinals that extend classical arithmetic
contain 0, then α is occasionally called a γ-number (see Additively indecomposable ordinal). These are exactly the ordinals of the form ωβ. The Cartesian
Ordinal_arithmetic
Prism with a 3-sided base
1161128. ISBN 1-59593-286-0. S2CID 3205780. Bagemihl, F. (1948). "On Indecomposable Polyhedra". American Mathematical Monthly. 55 (7): 411–413. doi:10.2307/2306130
Triangular_prism
Type of transfinite numbers
additively indecomposable ordinal) to be numbers γ > 0 such that α + γ = γ whenever α < γ, and delta numbers (see multiplicatively indecomposable ordinal)
Epsilon_number
Number of elements in a subset of a commutative group
simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups.[citation needed] Hence even the number of indecomposable summands of a group
Rank_of_an_abelian_group
Category with direct sums and certain types of kernels and cokernels
\mathbf {A} _{3}} identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of I is a full, additive
Abelian_category
Flat-sided three-dimensional shape
themathdoctors.org. Retrieved December 13, 2025. Bagemihl, F. (1948), "On indecomposable polyhedra", American Mathematical Monthly, 55 (7): 411–413, doi:10.2307/2306130
Polyhedron
Topics referred to by the same term
strike Profile ignition pickup, a term in automotive technology Proper indecomposable past set, a causal relation in a Lorentzian manifold Picture-in-picture
Pip
German mathematician (born 1945)
International Congress of Mathematicians in 1983 in Warsaw with talk Indecomposable representations of finite-dimensional algebras. He is a member of the
Claus_Michael_Ringel
In mathematics, element that equals its square
primitive idempotent of a ring R is a nonzero idempotent a such that aR is indecomposable as a right R-module; that is, such that aR is not a direct sum of two
Idempotent_(ring_theory)
in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either an automorphism
Fitting_lemma
Homomorphisms between simple modules over the same ring are isomorphisms or zero
properties are equivalent: M {\displaystyle M} is indecomposable; M {\displaystyle M} is strongly indecomposable; Every endomorphism of M {\displaystyle M} is
Schur's_lemma
Normed vector space that is complete
Hilbert space. An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional
Banach_space
Mathematical concept
inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1}. A right
Inverse_function
Type of module over a ring
1; this is a reformulation of the definition. Every simple module is indecomposable, but the converse is in general not true. Every simple module is cyclic
Simple_module
Algebraic structure
{\displaystyle m=n} and the degrees of the indecomposables in one decomposition are the same as the degrees of the indecomposables in the other decomposition (though
Polynomial_ring
British-Lebanese mathematician (1929–2019)
bundle is a sum of (essentially unique) indecomposable vector bundles, and then showing that the space of indecomposable vector bundles of given degree and
Michael_Atiyah
Mathematical ring with well-behaved ideals
over a left Noetherian module can be decomposed as a direct sum of indecomposable injective modules. See also #Implication on injective modules below
Noetherian_ring
Analysis of datasets using techniques from topology
{\displaystyle \rho _{s}^{t}} is the identity map. Interval modules are indecomposable. Although the result of Crawley-Boevey is a very powerful theorem, it
Topological_data_analysis
Representation theory of an important group in physics
space and time axis). However, there are finite-dimensional non-unitary indecomposable representations of the Poincaré algebra, which may be used for modelling
Representation theory of the Poincaré group
Representation_theory_of_the_Poincaré_group
Directed graph which is also a multigraph
isomorphism classes of indecomposable representations. Gabriel (1972) classified all quivers of finite type, and also their indecomposable representations.
Quiver_(mathematics)
Polyhedron with four faces
1093/oso/9780197640371.001.0001. ISBN 978-0-19-764037-1. Bagemihl, F. (1948). "On indecomposable polyhedra". American Mathematical Monthly. 55 (7): 411–413. doi:10.2307/2306130
Tetrahedron
Prime ideal that is an annihilator of a prime submodule
ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the spectrum S p e c ( R ) . {\displaystyle \mathrm
Associated_prime
Algebraic structure with addition and multiplication
the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition
Ring_(mathematics)
Topics referred to by the same term
(disambiguation) Compound (disambiguation) Decay (disambiguation) Food spoilage Indecomposability (disambiguation) All pages with titles containing Decomposition This
Decomposition (disambiguation)
Decomposition_(disambiguation)
R is a topological space whose points are (isomorphism classes of) indecomposable pure-injective right R-modules. Its closed subsets correspond to theories
Ziegler_spectrum
Integer side lengths of a right triangle
from the hypotenuse; that is, every primitive Pythagorean triangle is indecomposable. The set of all primitive Pythagorean triples forms a rooted ternary
Pythagorean_triple
group together with component-wise scalar multiplication. directing An indecomposable module M is directing if there is no cycle of module homomorphisms f
Glossary_of_module_theory
Probability distribution
of independent random variables whose probability distributions are indecomposable. Golomb coding is the optimal prefix code[clarification needed] for
Geometric_distribution
Manifold with inversion symmetry
{\displaystyle {\mathfrak {h}}} . The simple roots α1, ...., αn are the indecomposable positive roots. These can be numbered so that αi vanishes on the center
Hermitian_symmetric_space
Polish mathematician (1888–1945)
Mazurkiewicz and Hans Hahn. His 1935 paper Sur l'existence des continus indécomposables is generally considered the most elegant piece of work in point-set
Stefan_Mazurkiewicz
Ukrainian mathematician (born 1944)
for when commutative or noncommutative orders have only finitely many indecomposable lattices and helping classify hereditary and Bass orders. Together with
Yuriy_Drozd
Fraction with denominator a power of two
02.012, MR 2278053, S2CID 11566447 Nadler, S. B. Jr. (1973), "The indecomposability of the dyadic solenoid", The American Mathematical Monthly, 80 (6):
Dyadic_rational
Imputation (statistics) Incidence (epidemiology) Increasing process Indecomposable distribution Independence of irrelevant alternatives Independent component
List_of_statistics_articles
[e_{3},e_{1}]=e_{2};} g 4.1 {\displaystyle {\mathfrak {g}}_{4.1}} , indecomposable nilpotent, [ e 2 , e 4 ] = e 1 , [ e 3 , e 4 ] = e 2 ; {\displaystyle
Classification of low-dimensional real Lie algebras
Classification_of_low-dimensional_real_Lie_algebras
Generalization of "n-th" to infinite cases
multiplication of ordinals). Similarly, one can consider additively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly
Ordinal_number
Endomorphism algebra of an abelian group
non-trivial idempotent elements. If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring. For a semisimple
Endomorphism_ring
Decomposition method in algebra
case the ring R can be written as a direct sum R = e1R + ... + enR of indecomposable rings, which are sometimes also called the blocks of R. Albert, A. Adrian
Peirce_decomposition
(Mathematical) ring with a unique maximal ideal
the module M is local, then M is indecomposable; conversely, if the module M has finite length and is indecomposable, then its endomorphism ring is local
Local_ring
Branch of mathematics
appropriate category, and P consists of all isomorphism classes of indecomposable objects, i.e. objects which cannot be decomposed as a direct product
Abstract analytic number theory
Abstract_analytic_number_theory
modules are balanced modules. R is QF-2 if each indecomposable projective right module and each indecomposable projective left module has a unique minimal
Quasi-Frobenius_ring
Topics referred to by the same term
a gamma number may be: A value of the gamma function An additively indecomposable ordinal An ordinal Γα that is a fixed point of the Veblen hierarchy
Gamma_number
derived categories, many naturally occurring triangulated categories are "indecomposable". In particular, for a smooth projective variety X whose canonical bundle
Semiorthogonal_decomposition
Pure-injective modules in mathematics
compact as a Z-module. Together with the indecomposable finite modules over Z, this is a complete list of indecomposable algebraically compact modules. Many
Algebraically_compact_module
Can be continuously shrunk to a point
by Borsuk and Mazurkiewicz in their paper Sur les rétractes absolus indécomposables, C.R.. Acad. Sci. Paris 199 (1934), 110-112). There is some disagreement
Contractible_space
History of a branch of mathematics
One had the remarkable extensions of Clifford theory by Green to the indecomposable modules of group algebras. During this era, the field of computational
History_of_group_theory
Mathematical group that can be generated as the set of powers of a single element
representation theory transparent. In the positive characteristic case, the indecomposable representations of the cyclic group form a model and inductive basis
Cyclic_group
Classifies quivers of finite type in terms of Dynkin diagrams
isomorphism classes of indecomposable representations. Gabriel (1972) classified all quivers of finite type, and also their indecomposable representations.
Gabriel's_theorem
algebra of an algebra A is the endomorphism ring of the sum of the indecomposable modules of A. It was introduced by Auslander (1974). An Artin algebra
Auslander_algebra
Generalisation of Fourier transform to any ring
\bigoplus _{i}F[x]/(P_{i}(x)^{p^{s}})} . The modules occurring are now indecomposable rather than irreducible. Suppose p ∤ n {\displaystyle p\nmid n} so we
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
French mathematician (1933–2015)
category of quasi-coherent sheaves on X, by considering the spectrum of indecomposable injective objects in the category. This theorem, later vastly generalized
Pierre_Gabriel
Broadcast channel in information theory
Thomasian (1958). "Proof of shannon's transmission theorem for finite-state indecomposable channels". The Annals of Mathematical Statistics. 29 (4). United States:
Blackwell_channel
American mathematician
Society. ISBN 978-0-8218-0117-8. Bing, R. H. (1948). "A homogeneous indecomposable plane continuum". Duke Mathematical Journal. 15: 729–742. doi:10
R._H._Bing
Class of compact connected topological spaces
three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of an abelian compact topological group
Solenoid_(mathematics)
P=P_{1}\otimes P_{2}} where P1 is a probability distribution without any indecomposable factor and P2 is a distribution that is either degenerate or is representable
Khinchin's theorem on the factorization of distributions
Khinchin's_theorem_on_the_factorization_of_distributions
direct sum of multiples of finitely many indecomposables, which induce to a finite number of constituent indecomposable modules of which M is a direct sum.
Separable_algebra
Specialty in philosophy, focused on German language origin
"substantial forms of being" with the following properties: they are eternal, indecomposable, individual, subject to their own laws, un-interacting, and each reflecting
German_philosophy
When the linear combination of a random variable with itself has the same distribution
the maximum is available by a simple extension. Infinite divisibility Indecomposable distribution Lukacs, E. (1970) Section 5.7 Feller (1971), Section VI
Stability_(probability)
)} , then Δ ( γ ) {\displaystyle \Delta (\gamma )} is the set of all indecomposable elements of Φ + ( γ ) {\displaystyle \Phi ^{+}(\gamma )} root 1. root
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Theorem in group theory
where s ≥ 0, r ≥ 0, where each of the groups Ai is nontrivial, freely indecomposable (that is, it cannot be decomposed as a free product) and not infinite
Grushko_theorem
Topics referred to by the same term
Infra Project Tip and ring, two wires in a telephone line Terminal indecomposable past set Texas Instruments Power, a series of transistors Third-order
Tip
Method for constructing vector bundles
Horrocks (1964, section 10). His original construction gave an example of an indecomposable rank 2 vector bundle over 3-dimensional projective space, and generalizes
Horrocks_construction
{\displaystyle \operatorname {Hom} (V,W)=V^{*}\otimes W} . indecomposable An indecomposable representation is a representation that is not a direct sum
Glossary of representation theory
Glossary_of_representation_theory
American mathematician
with Bo Lian Liu: Brualdi, Richard A.; Liu, Bo Lian (1991). "Fully indecomposable exponents of primitive matrices". Proc. Amer. Math. Soc. 112 (4): 1193–1201
Richard_A._Brualdi
Complexity class used in circuit complexity
(1997-07-01). "Anti-Hadamard Matrices, Coin Weighing, Threshold Gates, and Indecomposable Hypergraphs". Journal of Combinatorial Theory, Series A. 79 (1): 133–160
TC0
Triangle with integer side lengths
coprime. There are infinitely many decomposable, and infinitely many indecomposable, primitive Heronian (non-Pythagorean) triangles with integer radii for
Integer_triangle
Lie algebra, usually infinite-dimensional
classify Kac–Moody algebras, it is enough to consider the case of an indecomposable matrix C, that is, assume that there is no decomposition of the set
Kac–Moody_algebra
INDECOMPOSABILITY
INDECOMPOSABILITY
INDECOMPOSABILITY
INDECOMPOSABILITY
Boy/Male
Hindu, Indian, Marathi, Russian
Inestimable
Female
English
English name derived from the vocabulary word, SERENITY means "serenity."
Girl/Female
Muslim
Powerful, Strong
Girl/Female
Indian
Fruitfulness, Plenty
Boy/Male
Bengali, Hindu, Indian, Modern
Brave
Girl/Female
Indian, Marathi, Tamil
Mullai Flower on a Vine
Boy/Male
Hindu, Indian, Sanskrit
Day and Night
Girl/Female
Indian
Boy/Male
Tamil
God knowledge
Girl/Female
Gujarati, Hindu, Indian
Dusk
INDECOMPOSABILITY
INDECOMPOSABILITY
INDECOMPOSABILITY
INDECOMPOSABILITY
INDECOMPOSABILITY