Definition:Assembly (Mathematical Theory)

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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:

  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 $\mathbf B$.

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

  1. 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