Kernel (set theory)

Summary

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.

DefinitionEdit

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]

QuotientsEdit

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 squareEdit

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 structuresEdit

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 topologyEdit

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 alsoEdit

ReferencesEdit

  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.

BibliographyEdit

  • Awodey, Steve (2010) [2006]. Category Theory. Oxford Logic Guides. Vol. 49 (2nd ed.). Oxford University Press. ISBN 978-0-19-923718-0.
  • Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.