# why algebraic structures?

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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Constant symbols can be viewed as $0$-ary function symbols, so they are included in the Wikipedia definition. And as far as relation symbols go, who cares about the Wikipedia definition? Of course ordered field is an algebraic structure. –  André Nicolas Jan 19 '13 at 17:27
Thanks for the insight. still my question holds regardless of the exact definition (I think :)). –  cruvadom Jan 19 '13 at 17:35
A set is obviously an algebraic structure: in fact, its signature is trivial. One reason for restricting attention to signatures without relation symbols is that there are fewer subtleties about them. For instance, every injective homomorphism between two algebraic structures of the same signature is an embedding, but this is not true if you have relation symbols. –  Zhen Lin Jan 19 '13 at 18:05
The way I look at it, a homomorphism is just the "non-relational structures" special case of the model theoretic definition for embedding: an injective function preserving all constants, functions and relations. –  cruvadom Jan 19 '13 at 18:12
@AndréNicolas An ordered field is certainly an algebraic structure in the colloquial sense, but not in the technical sense coming from universal algebra. –  Alex Kruckman Jan 19 '13 at 21:22

• As Zhen Lin states in the comments, algebraic structures are somewhat simpler to deal with than structures with both functions and relations. Restricting attention even further, a class of algebraic structures is called a variety of algebras if it can be defined using only equations (like $x\cdot (y\cdot z) = (x\cdot y) \cdot z$). Examples of varieties are groups, rings, modules, boolean algebras... Varieties are the core topic of study in the field of universal algebra, and they are very nice classes for a variety of reasons (e.g. they are closed under product and coproduct, they have free objects...). One could answer that we spend so much time studying and using varieties of algebras simply because they have lots of structure which is interesting and useful.