Definition:Isomorphism between Structures

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

• Definition:Isomorphic Structures

Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Definition $\text{II}.8.18$