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Suppose that two algebraic structures $(A,+, \cdot, ...)$ and $(B,+, \cdot, ...)$ are isomorphic (... in this case referring to any amount of n-ary operations on the sets A and B).

Besides the fact that we have a way to relate the products of any n elements in the source structure to the target structure and we know that they behave similarly with respect to their operations, what else can we always infer will remain the same between the structures? Do we know that certain theorems will hold in $(B,+, \cdot, ...)$ that have been shown to hold in $(A,+, \cdot, ...)$, or visa versa? Or is this generally contingent upon the structures in question?

An elaboration with an example would be greatly appreciated.

Thank you.

AnotherPerson
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2 Answers2

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Indeed. It is a fact from model theory that isomorphic structures are elementary equivalent. This fact is easy to prove by induction on the structure of the formula. Just to give you some example: If $G$ and $H$ are isomorphic groups and $G$ satisfies "every element of order $3$ is central", then the same holds in $H$.

Martin Brandenburg
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At least for statements expressible in a first order sentence there is the isomorphism lemma that states, that either both structures satisfy the sentence or both do not.

Sebastian Bechtel
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  • Does the order really matter? For example, the second order sentence "All sylow subgroups are normal" is transported by isomorphisms of groups, and I am pretty sure that you can transport any second order sentence for any algebraic theory. – Martin Brandenburg Apr 09 '15 at 22:03
  • I don't know whether or not it matters, I just know this statement for first order formulae. – Sebastian Bechtel Apr 10 '15 at 05:59