Irreducible (mathematics)

In mathematics, the term "irreducible" is used in several ways.

* In abstract algebra, irreducible can be an abbreviation for irreducible element; for example an irreducible polynomial.

* In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module.

* Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as "irreducible over an algebraic closure".

* In commutative algebra, a commutative ring "R" is irreducible if its prime spectrum, that is, the topological space Spec "R", is an irreducible topological space.

* A directed graph is irreducible if, given any two vertices, there exists a path from the first vertex to the second. A digraph is irreducible if and only if its adjacency matrix is irreducible.

* In a related notion, a matrix is irreducible if it is not similar to a block upper triangular matrix via a permutation. (Replacing non-zero entries in the matrix by one, and viewing the matrix as an adjacency matrix of a graph, the matrix is irreducible if and only if the graph is.)

* Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state.

* In the theory of manifolds, an "n"-manifold is irreducible if any embedded ("n"−1)-sphere bounds an embedded "n"-ball. Implicit in this definition is the use of a suitable category, 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 "n"-manifold is called prime, if it cannot be written as a connected sum of two "n"-manifolds (neither of which is an "n"-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 3-manifold 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 "S"¹ and the twisted 2-sphere bundle over "S"¹. See, for example, Prime decomposition (3-manifold).

* A topological space is irreducible if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much significance for Hausdorff spaces. See also irreducible component, algebraic variety.

* In universal algebra, irreducible can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible.

* A 3-manifold is P²-irreducible if it is irreducible and contains no 2-sided $mathbb\; RP^2$ (real projective plane).

* An Irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction.