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> I thought a a group, ring, etc. was precisely an example of a "universal algebra". No, this is incorrect. A specific group is not an example of an algebraic

> I thought a a group, ring, etc. was precisely an example of a "universal algebra". No, this is incorrect. A specific group is not an example of an algebraic structure. A specific group is an example of a group, and group structure is an example of an algebraic structure. > You seem to contradict yourself, at times agreeing with this statement ("even though one is an example of the other"), at times not ("a group is not an algebra"). The terminology is confusing, yes. I guess I didn't do a great job of explaining it. There are two things called "algebra" here and one is an example of the other, while also having examples of itself. I'll try to put things in more concrete terms. The real numbers, with addition and multiplication, is an example of an algebra. "Algebra" is an example of algebraic structure. "Group" is an example of algebraic structure. The real numbers, with addition and multiplication, is not an example of a group. As an analogy-- The apple I ate for lunch yesterday is an example of an apple. "Apple" is an example of a word. "Table" is an example of a word. The apple I ate for lunch yesterday is not an example of a word. > Edit: Wolfram Mathworld agrees with me: "Universal algebra studies common properties of all algebraic structures, including groups, rings, fields, lattices, etc." http://mathworld.wolfram.com/UniversalAlgebra.html http://mathworld.wolfram.com/UniversalAlgebra.html It does not agree. Note that we could add "algebras" to that list: "Including groups, rings, fields, lattices, algebras, etc." We are not talking about common properties of algebraic structures, we are talking about a specific algebraic structure, which is called "algebra." Universal algebra is a distinct topic from group theory and a distinct topic from linear algebra. The three fields are different studies. Yes, I know the terminology is confusing.

monoideism 7y ago
OK, I understand this meaning of "algebra" now. I've done some basic self-study of group theory, and of rings, but never really got to an algebra, which I see can be defined as "a ring that has the added structure of a field of scalars and a coherent multiplication". So it's a ring plus some added structure. I was unaware of this, which was adding to the confusion. It would have been helpful to differentiate that since my question/comment pertained to the term "universal algebra". I'm still not totally sure I understand what universal algebra studies now. Is it something akin to category theory?
- klodolph 7y ago
  > I'm still not totally sure I understand what universal algebra studies now. Is it something akin to category theory? Sort of, in the sense that it’s more general. You might study category theory, prove some theorems, and then apply those theorems to a specific category, like Grp (which is the category of groups). In category theory, you will prove the theory using “objects” and “morphisms” which are called “groups” and “homomorphisms” inside Grp. The same thing applies to universal algebra. You might prove some theorem in universal algebra, about algebraic structures in general, and then apply that proof to some specific algebraic structure, like algebras or groups. (And from a practical perspective… you might actually write the proof for a specific object first, and then e.g. generalize your group theory proof into a universal algebra proof.)

monoideism 7y ago

OK, I understand this meaning of "algebra" now. I've done some basic self-study of group theory, and of rings, but never really got to an algebra, which I see can be defined as "a ring that has the added structure of a field of scalars and a coherent multiplication". So it's a ring plus some added structure. I was unaware of this, which was adding to the confusion. It would have been helpful to differentiate that since my question/comment pertained to the term "universal algebra". I'm still not totally sure I understand what universal algebra studies now. Is it something akin to category theory?

klodolph 7y ago
> I'm still not totally sure I understand what universal algebra studies now. Is it something akin to category theory? Sort of, in the sense that it’s more general. You might study category theory, prove some theorems, and then apply those theorems to a specific category, like Grp (which is the category of groups). In category theory, you will prove the theory using “objects” and “morphisms” which are called “groups” and “homomorphisms” inside Grp. The same thing applies to universal algebra. You might prove some theorem in universal algebra, about algebraic structures in general, and then apply that proof to some specific algebraic structure, like algebras or groups. (And from a practical perspective… you might actually write the proof for a specific object first, and then e.g. generalize your group theory proof into a universal algebra proof.)