According to wikipedia, an algebraic structure is an arbitrary set with one or more finitary operations defined on it. From a model theory perspective, I understand this definition as: structure with no relations, only functions and constants (I added the constants to my definition, because I know that group, rings, have them). So for example an ordered field is not an algebraic structure (has relation), and a set isn't too (no function). Regardless of the exact definition, a massive part of mathematics is concerned with the study of these structures, and the relations between them. Geometry benefits from algebraic geometry, number theory from algebraic number theory, and so on. My question is, what do this sort of structures have in them, that makes us study and use them so much, and not other structures?
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This is not a complete answer, but I'm not sure that your question can be answered. So here are some thoughts: