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243:-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the
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223:, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. The notions of irreducibility in algebra and manifold theory are related. An
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topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over
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if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state.
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of coefficients. It applies in various situations, for example to irreducibility of a
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includes a list of related items that share the same name (or similar names).
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if it is not the union of two proper closed subsets. This notion is used in
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with no nontrivial proper subrepresentations. Similarly, an
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are smaller than those in any other equivalent fraction.
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385:Index of articles associated with the same name
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215: − 1)-sphere bounds an embedded
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40:if it cannot be factored over that field.
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316:if it is irreducible and contains no
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251:and the twisted 2-sphere bundle over
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362:(or fraction in lowest terms) is a
239:-manifolds (neither of which is an
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413:Set index articles on mathematics
344:{\displaystyle \mathbb {R} P^{2}}
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257:Prime decomposition (3-manifold)
231:, if it cannot be written as a
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145:irreducible topological space
181:.) A detailed definition is
154:is irreducible if it is not
51:can be an abbreviation for
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94:is a term applied to mean
73:irreducible representation
24:is used in several ways.
418:Mathematical terminology
303:subdirectly irreducible
167:upper triangular matrix
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135:is irreducible if its
92:Absolutely irreducible
84:is another name for a
61:irreducible polynomial
38:irreducible polynomial
353:real projective plane
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284:irreducible component
255:. See, for example,
108:linear representation
68:representation theory
360:irreducible fraction
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227:-manifold is called
116:irreducible over an
299:algebraic structure
126:commutative algebra
53:irreducible element
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272:algebraic geometry
179:strongly connected
82:irreducible module
389:set index article
295:universal algebra
288:algebraic variety
264:topological space
211:if any embedded (
199:In the theory of
118:algebraic closure
112:algebraic variety
98:, even after any
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280:Hausdorff spaces
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171:adjacency matrix
130:commutative ring
100:finite extension
76:is a nontrivial
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372:denominator
282:. See also
268:irreducible
209:irreducible
194:irreducible
160:permutation
110:, or of an
96:irreducible
49:irreducible
18:mathematics
407:Categories
310:3-manifold
245:3-manifold
183:given here
36:may be an
30:polynomial
368:numerator
201:manifolds
221:category
188:Also, a
143:, is an
318:2-sided
235:of two
156:similar
102:of the
32:over a
394:If an
158:via a
152:matrix
55:of an
387:This
229:prime
203:, an
173:of a
164:block
162:to a
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70:, an
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370:and
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358:An
312:is
293:In
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