There is a consensus, that "isomorphy" (the "is isomorphic to"-relation) is the right kind of sameness between universal algebras (say groups, (single-sorted) vector spaces, lattices, ...), because isomorphic objects share all their "algebraic properties". I wonder though:
Is there a (meta)theorem, that tells us exactly which properties, I can possibly come up with in the language of said algebraic structure, are indeed shared by isomorphic algebras?
If not, is there at least a general (meta)theorem stating, that a big class of (which?) properties are shared by isomorphic algebras?
This question might be related to model theory, although I have to say I know nothing about that.
Two isomorphic algebraic structures are elementarily equivalent, that is, both satisfy the same first-order sentences.
edit:
Initially when I answered this question I was under the impression that somehow isomorphic structures could disagree on second-order properties. Now, seeing the answers to this question and the comments to the present one more closely, I can see the absurd of isomorphic structures disagreeing on properties build up on their language no matter higher the underlying logic.
Anyway, I'll keep this answer for reference even if it's not a complete one.