I am studying Linear Algebra and often I come across things that belong the abstract algebra but still peak my interrest. I was reading up on metric and metric spaces (on wikipedia)
Wikipedia says that a metric space is an Ordered pair pair (M,d) where M is a set and d a metric
d: MxM—)$\mathbb{R}$. But how come a coordinate in the x-y plain is also a ordered pair (a,b) without a function b:AxA–) $\mathbb{R}$ (thats stupid I know but it try to make a point.)
And when I was learning about vectorspaces I came across a quadruple A vector space is defined as a quadruple $(\mathbf{V},\mathbb{K},\oplus,\odot)$ where $\mathbf{V}$ is a set of elements called vectors, $\mathbb{K}$ is a field $(\mathbb{K},+,\cdot)$ , $\oplus$ is a binary operation (called sum) on $\mathbf{V}$ such that $(\mathbf{V},\oplus)$ is an Abelian group and $a\odot\mathbf{v}:\mathbb{K}\times\mathbf{V} \rightarrow \mathbf{V}$ is a scalar multiplication. But how do you know the underling relation between these objects in the quadruple (ordererd pair(metric space)) Do all quadruples have the same formath of relations I mean does the first symbol and the third symbol always have a relation or does this need to be fined , another example is the second symbol always a field or can It be any object? Are the last 2 always "binaire operations" (yeah wikipedia says that scalar multplication can sometimes be called a binaire operations altough it clashes with the definition) Summary: I want to know what can be in these quadruples and How you can know the underling "relation" (probaly not the right word but I hope you get it) between these different things in the quadruple. I did some wikipedia surfing myself but didnt come to a clear answer and I dont realy know where to look.
Thanks in Advance
Best Answer
When we say $(A,\square,\triangle)$ is a field, we are saying that the base set of the field is $A$, that $\square$ and $\triangle$ are binary operations on $A$ satisfying certain properties.
We use an ordered triple because we need to distinguish between the first operation and second operation, they don't satisfy the same axioms. In the case of that field, for example, we know that $\triangle$ distributes over $\square$, but no the other way around, or that the every element of $A$ has an inverse with respect to the $\square$ operation, and that's not true for $\triangle$. If we write "let $A$ a field with operations $\#$ and $\&$" it's not clear from the statement which operation corresponds to the "addition" of field and which to the "multiplication". Writing them in a triple get rid of the ambiguity.
Also, I think you have a misconception, because you ask "how do you know the underling relation between these objects in the quadruple "(in this case, triple). The point is that the notation "$(A,\square,\triangle)$" alone is incomplete. The correct notation would be "$(A,\square,\triangle)$ is a field".
From $(x,y)$ we obviously can't know if that denotes a group or a metric space (or a completely different thing). Examples of COMPLETE notations are is "$(x,y)$ is a group" or "the metric space $(x,y)$".