# Definition:Bourbaki Assembly

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## Definition

The notion of **assembly** is an example of a collation used in 1968: Nicolas Bourbaki: *Theory of Sets*.

An **assembly** is a succession of signs written one after another.

Certain signs which are not variables are allowed to be joined in pairs by links, as follows:

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

### Link

In an **assembly**, certain signs which are not variables can be joined in (ordered) pairs by **links**, as follows:

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

where the $\overbrace {}{}$ depicts a **link** between $\tau$ and $\Box$.

### Construction of Assemblies

#### Concatenation

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

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

#### Extracting the Variable

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

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

- $(1): \quad$ Form the
**assembly**$\tau \mathbf A$ - $(2): \quad$ Link each occurrence of $x$ wherever it appears in $\mathbf A$ to the $\tau$ written to the left of $\mathbf A$
- $(3): \quad$ 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:

- $\paren {\mathbf B \mid x} \mathbf A$

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 $\paren {\mathbf B \mid x} \mathbf A$ is identical with $\mathbf A$.

In particular:

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

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

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### Notation

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.

## Also see

- Results about
**Bourbaki assemblies**can be found**here**.

## Sources

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