- #1

Rasalhague

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*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 [itex]f : F \times V \to V[/itex] 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 [itex]\lambda \in F[/itex] can be combined with every element [itex]\textbf{u} \in V[/itex]..."

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?