Kernel (set theory)

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In set theory, the kernel of a function (or equivalence kernel[1]) may be taken to be either

  • the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell",[2] or
  • the corresponding partition of the domain.

An unrelated notion is that of the kernel of a non-empty family of sets which by definition is the intersection of all its elements: This definition is used in the theory of filters to classify them as being free or principal.

Definition

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Kernel of a function

For the formal definition, let   be a function between two sets. Elements   are equivalent if   and   are equal, that is, are the same element of   The kernel of   is the equivalence relation thus defined.[2]

Kernel of a family of sets

The kernel of a family   of sets is[3]   The kernel of   is also sometimes denoted by   The kernel of the empty set,   is typically left undefined. A family is called fixed and is said to have non-empty intersection if its kernel is not empty.[3] A family is said to be free if it is not fixed; that is, if its kernel is the empty set.[3]

Quotients

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Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:  

This quotient set   is called the coimage of the function   and denoted   (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image,   specifically, the equivalence class of   in   (which is an element of  ) corresponds to   in   (which is an element of  ).

As a subset of the Cartesian product

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Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product   In this guise, the kernel may be denoted   (or a variation) and may be defined symbolically as[2]  

The study of the properties of this subset can shed light on  

Algebraic structures

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If   and   are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function   is a homomorphism, then   is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of   is a quotient of  [2] The bijection between the coimage and the image of   is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.

In topology

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If   is a continuous function between two topological spaces then the topological properties of   can shed light on the spaces   and   For example, if   is a Hausdorff space then   must be a closed set. Conversely, if   is a Hausdorff space and   is a closed set, then the coimage of   if given the quotient space topology, must also be a Hausdorff space.

A space is compact if and only if the kernel of every family of closed subsets having the finite intersection property (FIP) is non-empty;[4][5] said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.

See also

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References

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  1. ^ Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra, Chelsea Publishing Company, p. 33, ISBN 0821816462.
  2. ^ a b c d Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, vol. 301, CRC Press, pp. 14–16, ISBN 9781439851296.
  3. ^ a b c Dolecki & Mynard 2016, pp. 27–29, 33–35.
  4. ^ Munkres, James (2004). Topology. New Delhi: Prentice-Hall of India. p. 169. ISBN 978-81-203-2046-8.
  5. ^ A space is compact iff any family of closed sets having fip has non-empty intersection at PlanetMath.

Bibliography

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