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