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What type of widely accepted mathematical notations are available to access tuple components?

For exmaple, let suppose that I want to define a predicate $P(t,a)$ to be satisfied when the 3rd element of the 8-dimensional tuple $t$ is equal to $a$.
Are all of the following notations widely accepted?
Is there any additional widely accepted notation that I have not listed?

  1. $P(t,a) \equiv t[3]=a$
  2. $P(t,a) \equiv t(3)=a$
  3. $P(t,a) \equiv \exists b,c,e,f,g,h,i \in N:(b,c,a,e,f,g,h,i)=t$
  4. $P(t,a) \equiv t = (\ldots,\ldots,a,\ldots,\ldots,\ldots,\ldots,\ldots)$
  5. $P((b,c,d,e,f,g,h,i),a) \equiv t=d$

I also thought about using the following notation. First I define the operator $.third$ which look similar to object oriented programming syntax, then I use the operator in the definition of the predicate:

$\begin{align*} &\forall t=(b,c,d,e,f,g,h,i) \in N^8 \\ & \quad t.third \equiv d \\ & \\ &P(t,a) \equiv t.third = a \end{align*}$

Is the notation above correct and used anywhere?
If not, is there a way to refer to tuple parameters by "name" rather than index?

Robbo
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    One common notation is to just define $\pi_k : X_1 \times \cdots \times X_k \times \cdots \times X_n \to X_k$ defined by $\pi_k(x_1, \ldots, x_k, \ldots, x_n) = x_k.$ Often it can just be defined in text, saying "Let $\pi_k(t)$ give the $k$:th entry of $t$." – md2perpe Aug 28 '18 at 07:42
  • Thanks, Is there any notation that allows to "name" the position. I guest that I could define an indexing function $I$ from symbols to index and then define $I(third) \equiv 3$. Then I could use $\pi_{I(third)}(t)$. I writing a computer science paper so I get often to use tuples as objects and becomes handy to be able to refer to a component via a name – Robbo Aug 28 '18 at 08:39
  • I would then suggest something like $\operatorname{third}(t) = t_3$ if $t = (t_1, \ldots, t_n).$ – md2perpe Aug 28 '18 at 08:56
  • Thanks. Is there any book on standard mathematical notation? This may be another question I guess though. – Robbo Aug 28 '18 at 09:07

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