# Definition:Assembly (Mathematical Theory)

## Definition

An assembly in a mathematical theory is a succession of signs written one after another, along with other delineating marks according to the specific nature of the theory under consideration.

## Bourbaki Definition

The mathematical theory as defined in 1968: Nicolas Bourbaki: Theory of Sets allows certain signs which are not letters to be joined in pairs by links, as follows:

$\overbrace {\tau A \Box}^{} A'$

### Construction of Assemblies

#### Concatenation

Let $\mathbf A$ and $\mathbf B$ denote assemblies.

Then $\mathbf {AB}$ denotes the assembly formed from writing the assembly $\mathbf B$ immediately to the right of assembly $\mathbf A$.

#### Extracting the Variable

(or whatever this operation is called)

Let $\mathbf A$ denote an assembly and let $x$ be a letter.

Then the assembly $\map {\tau_x} {\mathbf A}$ is constructed as follows:

1. Form the assembly $\tau \mathbf A$
2. Link each occurrence of $x$ wherever it appears in $\mathbf A$ to the $\tau$ written to the left of $\mathbf A$
3. Replace $x$ by $\Box$ wherever it occurs in $\mathbf A$.

The assembly denoted $\map {\tau_x} {\mathbf A}$ therefore does not contain $x$.

#### Replacement

Let $\mathbf A$ and $\mathbf B$ denote assemblies.

We denote by:

$\paren {\mathbf B | x} \mathbf A$

the assembly obtained by replacing $x$, wherever it occurs, by $\mathbf B$.

$\mathbf B$ replaces $x$ in $\mathbf A$.
If $x$ does not appear in $\mathbf A$, then $\paren {\mathbf B | x} \mathbf A$ is identical with $\mathbf A$.
$\paren {\mathbf B | x} \map {\tau_x} {\mathbf A}$
is identical with $\map {\tau_x} {\mathbf A}$.