Kernel

Definition

Given a function between two spaces $X$ and $Y$ of the same type of space (which is imbued with an identity element), the kernel of $f:X\rightarrow Y$ (where $f$ is a function) is defined as:

• $$\text{Ker}(f)=\{x\in X|f(x)=e\}$$ where $$e$$ denotes the identity of $Y$

Potential generalisation

Given a function $$f:X\rightarrow Y$$ the kernel where $Y$ is a space imbued with the concept of identity (this identity shall be denoted $e$) written as follows:

• $$\text{Ker}(f)=\{x\in X|f(x)=e\}$$

Vector spaces

Given a linear map $T\in\mathcal{L}(V,W)$ where $V$ and $W$ are vector spaces over the field $F$, that is $T:U\rightarrow V$, the kernel of $T$ is:

• $$\text{Ker}(T)=\{v\in V|Tv=0\}$$ using the notation $$Tv=T(v)$$

Potential generalisation

Given any function $f:X\rightarrow V$ where $X$ is any set and $V$ a vector space, we may define the kernel of $f$ as follows:

• $$\text{Ker}(f)=\{x\in X|f(x)=0\}$$ where $$0$$ denotes the additive identity of the vector space

Groups

Given a homomorphism $f:(G,\times_1)\rightarrow(H,\times_2)$ between two groups, $G$ and $H$, we define the kernel of $f$ as follows:

• $$\text{Ker}(f)=\{g\in G|f(g)=e_H\}$$ where $$e_H$$ is the identity of $$H$$

Potential generalisation

I believe we can extend this definition to any map:

Given a group $(G,\times)$ where $e$ denotes the identity, and a function $f:X\rightarrow G$ where $X$ is any set. The :kernel is defined as follows:

• $$\text{Ker}(f)=\{x\in X|f(x)=e\}$$