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.

Definition

For the formal definition, let X and Y be sets and let f be a function from X to Y. Elements x₁ and x₂ of X are equivalent if f(x₁) and f(x₂) are equal, i.e. are the same element of Y. The kernel of f is the equivalence relation thus defined.[2]

Quotients

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

\left\{\,\left\{\,w\in X\mid f(x)=f(w)\,\right\}\mid x\in X\,\right\}.

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

As a subset of the square

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

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

.[2]

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

In 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 from 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. See also Kernel (algebra).

In topological spaces

If X and Y are topological spaces and f is a continuous function between them, 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.

References

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

Sources

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

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[mac-lane-birkhoff-1] Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra, Chelsea Publishing Company, p. 33, ISBN 0821816462.

[bergman-2] Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, 301, CRC Press, pp. 14–16, ISBN 9781439851296.