Kernel (set theory)

In set theory, the kernel of a function f (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 f 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 \mathcal{B}, which by definition is the intersection of all its elements:

\ker \mathcal{B} ~=~ \bigcap_{B \in \mathcal{B}} \, B.

This definition is used in the theory of filters to classify them as being free or principal.

Definition

'

For the formal definition, let f : X \to Y be a function between two sets. Elements x_1, x_2 \in X are equivalent if and only if f\left(x_1\right) and f\left(x_2\right) are equal, that is, are the same element of Y. The kernel of f is the equivalence relation thus defined.^[2]

'

The is^[3]

\ker \mathcal{B} ~:=~ \bigcap_{B \in \mathcal{B}} B.

The kernel of \mathcal{B} is also sometimes denoted by \cap \mathcal{B}. The kernel of the empty set, \ker \varnothing, is typically left undefined. A family is called and is said to have if its kernel is not empty.^[3] A family is said to be if it is not fixed; that is, if its kernel is the empty set.^[3]

Quotients

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

\left\{\, \{w \in X : f(x) = f(w)\} ~:~ x \in X \,\right\} ~=~ \left\{f^{-1}(y) ~:~ y \in f(X)\right\}.

This quotient set X /=_f is called the coimage of the function f, and denoted \operatorname{coim} f (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, \operatorname{im} f; specifically, the equivalence class of x in X (which is an element of \operatorname{coim} f) corresponds to f(x) in Y (which is an element of \operatorname{im} f).

As a subset of the Cartesian product

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

\ker f := \{(x,x') : f(x) = f(x')\}.

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

Algebraic structures

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

In topology

If f : X \to Y is a continuous function between two topological spaces then the topological properties of \ker f can shed light on the spaces X and Y. For example, if Y is a Hausdorff space then \ker f must be a closed set. Conversely, if X is a Hausdorff space and \ker f is a closed set, then the coimage of f, 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.

References

1. ^ Mac Lane, Saunders & Birkhoff, Garrett (1999), Algebra, Chelsea Publishing Company, p. 33, ISBN 0821816462.
2. ^ Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Vol. 301, Pure and Applied Mathematics, CRC Press, pp. 14–16, ISBN 9781439851296.
3. ^ 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. ^

Bibliography

• Awodey, Steve (2010 [2006]). Category Theory. Vol. 49. Oxford Logic Guides. 2nd ed. Oxford University Press. ISBN 978-0-19-923718-0.