# 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:

- $\overbrace {\tau A \Box}^{} A'$
^{[1]}

### 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:

- Form the assembly $\tau \mathbf A$
- Link each occurrence of $x$ wherever it appears in $\mathbf A$ to the $\tau$ written to the left of $\mathbf A$
- 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 $\mathbf B$.

This is read::

**$\mathbf B$ replaces $x$ in $\mathbf A$.**

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

In particular:

- $\paren {\mathbf B | x} \map {\tau_x} {\mathbf A}$

is identical with $\map {\tau_x} {\mathbf A}$.

## Notes

- ↑ The specific symbology used by Bourbaki in
*Theory of Sets*has not been rendered accurately here, as the author of this page has not been able to establish a method by which to do it. However, the intent has been expressed as accurately as possible.

## Sources

- 1968: Nicolas Bourbaki:
*Theory of Sets*... (previous) ... (next): Chapter $\text I$: Description of Formal Mathematics: $1$. Terms and Relations: $1$. Signs and Assemblies