Definition:Isomorphism between Structures
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Definition
Let $\AA, \BB$ be structures for a signature $\LL$.
Then an isomorphism between $\AA$ and $\BB$ is a bijection $\Phi: A \to B$ such that:
- $A$ and $B$ are the respective underlying sets of $\AA$ and $\BB$
- For each function symbol $f$ of arity $n$, we have, for all $a_1, \ldots, a_n \in A$:
- $\map {f_\BB} { \map \Phi {a_1}, \ldots, \map \Phi {a_n} } = \map \Phi { \map {f_\AA} { a_1, \ldots, a_n } }$
- For each predicate symbol $p$ of arity $n$, we have, for all $a_1, \ldots, a_n \in A$:
- $\map {p_\BB} { \map \Phi {a_1}, \ldots, \map \Phi {a_n} } = \map {p_\AA} { a_1, \ldots, a_n }$
- Note that in particular, for $n = 0$, this reduces to $f_\BB = \map \Phi {f_\AA}$ and $p_\BB = p_\AA$
Also see
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Definition $\text{II}.8.18$