Functionary structures

Many mathematical strucutres arise by defining a tupel with an abstract Set X and a function f on this set that satisfies some properties.

(X,\textcolor{lightblue}{f}) \quad X \ abstract \ set \qquad \textcolor{lightblue}{f} \ function

Properties

Lets overview the different properties of functions used to define new structures.

Codomain-Valued

Most structures need a function that has a specific codomain. The three most dominant classes are:

\begin{align*} &\textcolor{lightblue}{f} : X \to \textcolor{Maroon}{\digamma} \qquad codomain: ordered \ field \quad (\textcolor{Maroon}{\digamma},\leq) \\ & \textcolor{lightblue}{f} : X \to \textcolor{brown}{\mathbb{R}} \qquad codomain: real \ numbers \\ &\textcolor{lightblue}{f}:X \to \textcolor{yellow}{\mathbb{C}} \qquad codomain : complex \ numbers\end{align*}

Symmetric

A function is said to be symmetric if any of the following:

\forall \ x,y \in X \hspace{2cm}\textcolor{lightblue}{f}(x,y) = \textcolor{lightblue}{f}(y,x)

Conjugated Symmetric

A function is said to be conjugated symmetric if any of the following:

\forall \ x,y \in X \hspace{2cm}\textcolor{lightblue}{f}(x,y) = \overline{\textcolor{lightblue}{f}(y,x)}

Subadditive

A function is said to be subadditive if any of the following:

\forall \ x,y \in X \hspace{2cm} \textcolor{lightblue}{f}(x,y) \leq_{\textcolor{Maroon}{\digamma}} \textcolor{lightblue}{f}(x,z) + \textcolor{lightblue}{f}(z,y)

Semidefinite

A function is said to be semidefinite if any of the following:

\forall \ x,y\in  X \hspace{2cm} x = y \quad \implies \quad \textcolor{lightblue}{f}(x,y) = 0

Definite

A function is said to be definite if any of the following:

\forall \ x,y\in  X \hspace{2cm} x = y \quad \iff \quad \textcolor{lightblue}{f}(x,y) = 0

Positive

A function is said to be positive if any of the following:

\forall \ x \in  X \hspace{2cm} \textcolor{lightblue}{f}(x) \in \textcolor{Maroon}{\digamma}^+ 
\forall \ x \in X \hspace{2cm} 0 \leq_{ \textcolor{Maroon}{\digamma}} \textcolor{lightblue}{f}(x)

Compatible

There are different forms of being compatible. One structure often encountered is the vector space structure and we call a function that is compatible with this structure homogen, absolute homogen, additive or linear. Therfor let X be a vector space over a field F. The we have the following properties:

Homogen
\forall \ x \in X \ \forall k \in F : \qquad \textcolor{lightblue}{f}(k \cdot x) = k \cdot \textcolor{lightblue}{f}(x)
Absolute Homogen
\forall \ x \in X \quad  \forall \ k \in F : \qquad \textcolor{lightblue}{f}(k \cdot x) = |k| \cdot \textcolor{lightblue}{f}(x)
Additive
\forall \ x,y \in X: \qquad \textcolor{lightblue}{f}(x+y) = \textcolor{lightblue}{f}(x) + \textcolor{lightblue}{f}(y)
F-Linear
\forall \ x,y \in X  \quad \forall \ k \in F  : \qquad \textcolor{lightblue}{f}(k \cdot x+y) = k \cdot  \textcolor{lightblue}{f}(x) + \textcolor{lightblue}{f}(y)