How might we formally define the concatenation of two strings?

Question

Below we offer some definitions of string.

How would you mathematically define the concatenation of strings?

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The $\mathtt{HELLO\ WORLD}$ Example

$“\mathtt{HELLO}” + “\mathtt{\ }” + \mathtt{WORLD}” = “\mathtt{HELLO\ WORLD}”$

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The $\mathtt{HOW\ ARE \ YOU}$ Example

$\mathbb{WS} = \{ “\mathtt{HOW}”, “\mathtt{ARE}”, “\mathtt{YOU}” \}$

$\sum_{\sigma \in \mathbb{WS}}^{\text{}} \sigma = “\mathtt{HOW\ ARE \ YOU}”$

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Definitions of Unicode String

Fuzzy Definition of Unicode String

$\require{enclose}$ $\require{cancel}$ A string is function whose inputs are numbers and the outputs are Unicode characters such that indexing begins at $1$, and we have $\forall k \in \mathbb{Z} \setminus \{0\} \enclose{horizontalstrike}{\sigma(-|k|) = \sigma(+|k|)}$

Edit

I meant to write $\sigma(n-|k|) = \sigma(|k|)$ where $n$ is the minimum non-negative zalen integer from $\mathbb{Z}$ such that $\sigma(n-|k|) = \mathtt{null}$

Detailed Definition of Unicode String

For any mapping $\sigma$ from the zalen integers $\mathbb{Z}$ without zero to the set of all Unicode character $\mathbb{U}$ we say that $\sigma$ is a string

if and only if

There exists $n$ element of the zalen without zero such that:

if

the absolute value of $n$ is less than or equal to the absolute value of $m$
then
for any $k$ element of the set of all zalen without zero,

all of the following:

if

$-|n|$ is less than or equal to $k$
and
$k$ is less than or equal to the absolute value of $n$
then
$\sigma(k)$ not equal to $\mathtt{null}$

if

$k$ strictly less than $-|n|$
or
$k$ strictly greater than $+|n|$
then
$\sigma(k) = \mathtt{null}$

Additionally, $\forall n \in \mathbb{Z} \setminus \{0\}, \sigma(-|n|) = \sigma(+|n|)$

Mostly Symbolic Representation of the Definition of Unicode String

$\forall \sigma \subseteq (\mathbb{Z} \setminus \{0\}) \times (\mathbb{U}) \in $ the set of all finite-length strings

$\iff$

$\forall z, z^{\prime} \in \mathbb{Z} \setminus \{0\} \land \forall \mathtt{ch}, \mathtt{ch}^{\prime} \in \mathbb{U} (\sigma(z) = \sigma(z^{\prime})) \implies (\mathtt{ch} = \mathtt{ch}^{\prime})$

$\land$

$\exists n \in \mathbb{Z} \setminus \{0\} \backepsilon$

$|n| \leq |m|$
$\implies$
$\forall k \in \mathbb{Z} \setminus \{0\}$,

all of the following:

$-|n| \leq k \leq +|n|$
$\implies$
$\sigma(k) \neq \mathtt{null}$

$k < -|n|$ or $k > +|n|$
$\implies$
$\sigma(k) = \mathtt{null}$

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Mostly Wordy Definition of Subsequence

The question was, how would you define concatenation of strings.

However, it might be easiest to copy, paste, and edit a definition for subsequence.

For any two strings $\sigma$ and $\Sigma$, $\sigma$ is a sub-sequence of $\Sigma$ if and only if for every $z$ element of the zalen integers there exists $K$ element of the zalen integers such that $k$ is less than or equal to $K$ and $\sigma(k)$ is $\Sigma(K)$

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Mostly symbolic Definition of Subsequence

The question was, how would you define concatenation of strings.

However, it might be easiest to copy, paste, and edit a definition for subsequence.

For any two strings $\sigma$ and $\Sigma$, $\sigma$ is a sub-sequence of $\Sigma \iff \forall z \in \mathbb{Z} \setminus \{0\} \exists K \in \mathbb{Z} \setminus \{0\}: k \leq K \land \sigma(k) = \Sigma(K)$

Answer 1

If $\Sigma$ is a finite set of alphabets then we can define a string recursively as:

1. $\lambda$ (the empty string) is a string;
2. $\forall\; x \in \Sigma$, $x$ is a string;
3. If $a$ and $b$ are strings then so is $ab$.

You can add the white-space character(s) in $\Sigma$ if you wish.