Gödelization
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A Gödelization or Gödel numbering of a $\Sigma$-algebra $A$ is a pair $(C,\beta)$ with $C$ as a recursive $\Sigma$-number-algebra and $\beta\colon A \longrightarrow C$ as a $\Sigma$-algebra-morphism from $A$ onto $C$. As usual, $\Sigma$ designates a signature. A Gödelization could be understood as encoding of the elements of $A$ by natural numbers .
Gödelizations with bijective mapping $\beta$ are of special interest. They are the counterpart of a certain bijective coordinatisation mapping $\nu\colon C \longrightarrow A$. More precisely, $\beta=\nu^{-1}$ is the inverse of $\nu$ in this case. For a sensible signature $\Sigma$ and a term-generated $\Sigma$-algebra $A$, such a coordinatization with bijective mapping $\nu$ always exists. Taking together, every Gödelization induces canonically a corresponding coordinatization and vice versa.
The correspondence between Gödelization and coordinatization can be extended to congruences defined on $A$ and $C$. A congruence $\sim^\beta \subseteq A \times A$ induces a congruence $\sim^\nu \subseteq C \times C$ and vice versa via $\beta$ resp. $\nu$ .
As one can easily see, a Gödelization with bijective mapping $\beta$ does not always exist. As a simple counterexample take a $\Sigma$-algebra $A$ with an uncountable number of elements.
References
| [1] | M. Wirsing: "Algebraic Specification", in J. van Leeuwen: "Handbook of Theoretical Computer Science", Elsevier 1990 |
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