An algebraic structure on a set A (called the underlying set, carrier set or domain) is a collection of operations on A of finite arity, together with a finite set of identities, called axioms of the structure that these operations must satisfy. In the context of universal algebra, the set A with this structure is called an algebra,[1] while, in other contexts, it is (somewhat ambiguously) called an algebraic structure, the term algebra being reserved for specific algebraic structures that are vector spaces over a field or modules over a commutative ring.
Examples of algebraic structures include groups, rings, fields, and lattices.
I’m going to agree with knzhou here. Unfortunately, the terminology in mathematics can be misleading.
It is important to note context, and note the part where the article you quoted uses the works “ambiguously”, because the word “algebra” has more than one meaning.
In this case, a group is not an algebra (because we are talking in the context of algebras over a field or ring, not universal algebras).
It is unfortunate that the words are defined this way, but you have to deal with it. A “universal algebra” is a very different concept from an “algebra” (over a ring or field) even though one is an example of the other.
It’s like saying that “book” is a very different concept from “The Great Gatsby”.
Huh? I thought a a group, ring, etc. was precisely an example of a "universal algebra". 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").
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
The word "algebra" means a number of different things.
It's the name of a whole field in mathematics, which covers objects like groups, rings, fields, and so forth. The sort of things defined, very handwavily, in terms of operations you can do with their elements and the equations they satisfy.
It's the name of a rather specific kind of mathematical structure: roughly, a vector space together with a way of multiplying its elements. (Motivating example: n-by-n matrices.)
It's the name (but usually with some qualifiers to make it clearer what you mean) for a broad range of mathematical structures, of which the one in the previous paragraph is a special case. See e.g. https://en.wikipedia.org/wiki/F-algebra.
Groups are among the things studied in the field of algebra. Groups aren't algebras in the second, specific, sense. They are F-algebras. Most mathematicians, most of the time, would not call groups "algebras" without some qualifier like that F- prefix.
The term "Clifford algebra" also means some different things.
A Clifford algebra is a particular sort of algebra-in-the-second-sense. The field called Clifford algebra is the study of those things. These days people more often say "geometric algebra" rather than "Clifford algebra" for that meaning.
There are mathematical objects called Clifford groups. They are not at all the same thing as Clifford algebras, and you can study Clifford algebras in some depth without paying any attention to the Clifford groups. But they are closely related to the Clifford algebras.
Both Clifford groups and Clifford algebras have applications in quantum physics and, more specifically, in quantum computing. But so far as I know the ways in which you use them in quantum computing has very little to do with the ways in which you use Clifford algebras for doing geometry. It is quite common in mathematics for the same (or equivalent) objects and structures to turn up in multiple places, in unrelated-looking ways. Sometimes this gives rise to deep connections between different fields; sometimes it's just a coincidence. I don't know enough about either quantum computation or geometric algebra to know which of those is going on here, but my intuition leans toward "coincidence".
So. vtomole's original comment was kinda-right and kinda-wrong: yes, there is a connection between Clifford algebras and quantum computation, but it doesn't have much to do with the stuff discussed, e.g., at the far end of the top-level link here. knzhou's question was a good one, pointing out that the two topics are quite separate. vtomole's reply "A group is an algebraic structure ..." didn't make any untrue statements but did miss the point; the fact that a group is an algebraic structure doesn't mean that something called "the X group" necessarily has anything to do with something called "the X algebra" -- though it happens that in this case there is a connection. (vtomole clearly got the point soon after, as seen from their subsequent replies.) knzhou was correct to point out that vtomole's reply missed the point. lisper, again, didn't say anything untrue but I think he missed the point. klodolph's comment about algebras versus universal algebras versus algebra was spot-on and the only reason why I went into more detail above is that it was apparently too brief to be clearly understood. monoideism is right that (e.g.) a group is a universal algebra, wrong to say that klodolph is self-contradictory, and I think missing the point that "algebra" is used with different meanings on different occasions, and in the phrase "Clifford algebra" the specific meaning in question is not "universal algebra". (Even though a Clifford algebra is, also, an example of a universal algebra.)
The fact that a group is a universal algebra doesn't at all licence any sort of blurring of the distinction between Clifford groups and Clifford algebras. The meaning of "algebra" in "Clifford algebra" is not "universal algebra" or "F-algebra", it is "vector space with multiplication", and a group simply isn't one of those (well, some groups are, but e.g. the Clifford groups are not).
> 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
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.
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?
> 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.)
