Defining algebraic structures as n-tuples

[Rasalhague]

Bowen & Wang, in Introduction to Vectors and Tensors often define some algebraic structure as an n-tuple, e.g. "A semigroup is a pair (G,*) consisting of a nonsempty set G with an associative binary operation: (a*b)*c=a*(b*c) for all a,b,c in G." For more complicated structures, they use this kind of definition repeatedly: "A vector space is a 3-tuple (V,F,f) consisting of an additive abelian group V, a field F and a function $f : F \times V \to V$ such that..."

What's niggling me is that, having gone to such lengths to define these structures precisely in terms of sets, the distinction is then blurred between elements of these tuples and elements of their elements, and so on, e.g. "A vector space V is a 3-tuple (V,F,f) consisting of an abelian group V, a field F, and an operation f, called scalar multiplication, in which every scalar $\lambda \in F$ can be combined with every element $\textbf{u} \in V$..."

The symbol V is used three times in this definition, each time with a different referent: (1) vector space in their n-tuple sense, (2) abelian group in their n-tuple sense, (3) set of vectors on which the abelian group is defined!

But more often I read about vectors as "elements of a vector space", meaning elements of the third V, the set of vectors, rather than elements of the 3-tuple which Bowen & Wang define as a vector space. The more common usage seems to be to define a set as, say, a group G with some operation, or a set M as a manifold with some topology and atlas of coordinate charts. Is there a generally recognised name or notation for the set over which an algebraic structure is defined which distinguishes it from the n-tuple that is the structure itself?

I suppose the common way of describing these things is equivalent to Bowen & Wang's more formal definition in terms of n-tuples, but the name that Bowen & Wang give to the whole structure is more often given to just a part of it, a certain set. Is that right? Maybe I could call a set of vectors "the vector space V", and the n-tuple (V,F,f) the "(vector space) structure of V". Is that the usual practice, or are there more standard terms?

 

[Hurkyl]

One standard term is "underlying set". For example, if (G,*) is a group, then G is its underlying set.

Continuing the example, another term I think is fairly standard is that (G,*) is a "group structure" on G.




The purpose of this is two-fold:

1. It says "hey, we can reduce things to set theory"
2. It gives a convenient way to name the group operation when desired -- e.g. an easy way to disambiguate if we are working both with the semigroup (C,+) and the semigroup (C,*)

I don't think I've ever seen it used in any other way. (in particular, we pretty much never ever actually use it as an ordered tuple)

 

[Rasalhague]

One standard term is "underlying set". For example, if (G,*) is a group, then G is its underlying set.

Continuing the example, another term I think is fairly standard is that (G,*) is a "group structure" on G.

Thanks Hurkyl! So the pair (G,*) is the group itself and, synonymously, a group structure on its underlying set G. And if I'm reading something that refers to a Euclidean vector space, the set of real n-tuples, $\mathbb{R}^{n}$, is the underlying set of this vector space, and vector space itself is the 3-tuple $(\mathbb{R}^{n}, \mathbb{R},*)$, where * is scalar multiplication, and this 3-tuple is also called the vector space structure on the underlying set $\mathbb{R}^{n}$, the vectors of this vector space being the elements of its underlying set? And when people call a vector an element of a vector space, they're talking in a loose, informal way, and the more correct (if wordier) statement is that a vector is an element of the underlying set on which the vector space is defined?

The purpose of this is two-fold:

1. It says "hey, we can reduce things to set theory"
2. It gives a convenient way to name the group operation when desired -- e.g. an easy way to disambiguate if we are working both with the semigroup (C,+) and the semigroup (C,*)

Yeah, it seems like a useful idea.

I don't think I've ever seen it used in any other way. (in particular, we pretty much never ever actually use it as an ordered tuple)

Oh, maybe I misunderstood... What did you mean by the notation (G,*) if not a tuple? Bowen & Wang use that notation for a tuple, defined as an ordered set.

 

[Hurkyl]

And when people call a vector an element of a vector space, they're talking in a loose, informal way, and the more correct (if wordier) statement is that a vector is an element of the underlying set on which the vector space is defined?

If you are taking the syntactic position that everything is a set, and "an element of" always refers to the set membership relation, then you are correct.

Oh, maybe I misunderstood... What did you mean by the notation (G,*) if not a tuple? Bowen & Wang use that notation for a tuple, defined as an ordered set.

I did mean it's a tuple. But I meant in an intuitive sense that we don't really use it in a tuple-like way.

But that's just my intuition. If it doesn't appeal to you, then ignore that comment.

 

[Rasalhague]

Well, I'm relieved to have these terms "underlying set" and "group structure"; at least now I can distinguish if need be. I wonder if some people rigorously identify "group" with the underlying set, rather than the structure; it does seem pretty universal to talk about the elements of a vector space, etc.

Carol Whitehead, in Guide to Abstract Algebra, writes:

Let G be a nonempty set on which a binary operation * is defined [...] [Axioms.] [...]Then G is called a group with respect to the binary operation * and denoted by (G,*).

But I can see how that might be awkward, since according to her definition, G = (G,*), and also G = (G,+), but (G,*) might not = (G,+), and therefore G might not equal G...