What would be a good formal definition for the set $\mathcal T$ constructed below? I will try and give a simple example.
Consider the set $[\;n\;]=\{1,2,\ldots,n\}$, as well the family $\mathcal{C}$ of all subsets of cardinality 3 of $[\;n\;]$: $$\mathcal{C}=\{\{1,2,3\}, \{1,2,4\}, \ldots, \{n-2,n-1,n\}\}.$$ Obviously, every set $C \in \mathcal{C}$ has the same structure - it consists of three elements, $C=\{c^{C}_{1},c^{C}_{2},c^{C}_{3}\}$ with $c^{C}_{1}<c^{C}_{2}<c^{C}_{3}$ - or in other words, every set $C \in \mathcal{C}$ has a smallest, a second-smallest and a largest element.
I would now like to define a set $\mathcal{T}=\{t_{1},t_{2},t_{3}\}$ which generalizes the sets $C\in \mathcal{C}$ in the following sense: $t_{1}$ is a symbol representing the smallest element of a three-element set, $t_{2}$ represents the second-smallest element, and $t_{3}$ represents the largest element. (The intention is to view $\mathcal{T}$ as a poset and use its properties to draw conclusions about $\mathcal{C}$, but that is irrelevant for the current question.)
One possible way of defining $\mathcal{T}$ might be to construct its elements as equivalence classes in the following way:
Definition Ver. 1: Consider a multiset $$D=\bigcup_{C\in\mathcal{C}}\bigcup_{j=1}^{3}\left\{ c_{j}^{C}\right\}$$ and the following equivalence relation $\sim$ defined on $D$: $$c_{j_{1}}^{C_{1}}\sim c_{j_{2}}^{C_{2}}\Longleftrightarrow j_{1}=j_{2}.$$ We define the set $\mathcal{T}$ as $\mathcal{T}=D/\sim$ and its elements as the equivalence classes $t_{j}=\left[c_{j}^{C}\right]$.
What I like about this version is the idea of defining each $t_{j}$ as an equivalence class, which gets rid of this weird "$t_{j}$ is a symbol representing the $j^\text{th}$-smallest element of a set" formulation. On the other hand - and this is my biggest issue here - is this equivalence relation well-defined? The statement $j_{1}=j_{2}$ makes sense while we're talking about the variables $c_{j_{1}}^{C_{1}}$ and $c_{j_{2}}^{C_{2}}$; but I'm somewhat inclined towards looking at them as their values, i.e. the actual elements of the set $[\;n\;]$, at which point they lose their information about which index $j$ and subset $C$ they are associated with... Which interpretation is the correct one here?
Here's an alternative version for this definition:
Definition Ver. 2: Consider the set $$\mathcal{T} = \bigcup_{j=1}^{3}\left\{ t_{j}\right\}.$$ For a given set $C \in \mathcal{C}$, identify every element $c^{C}_{j}$ of $C$ with the element $t_{j}$ of $\mathcal{T}$ via the isomorphism $\tau_{C}:\mathcal{T}\rightarrow C$, $\tau_{C}:t_{j}\mapsto c^{C}_{j}.$
Using isomorphisms kind of makes more sense to me. On the other hand, now I'm stuck with my elements $t_{j}$ of $\mathcal{T}$ just being symbols without any further meaning to begin with. Furthermore, it seems like a strange construction to identify elements $t_{j}$ and $c^{C}_{j}$ by matching their indices $j$ - is it even clear when writing it like this? And is it legitimate to define $\mathcal{T}$ and $\tau_{C}$ together in one breath like that? (I could separate the definitions, but that makes the introduction of the object $\mathcal{T}$ even less understandable...)
Any help to clarify this will be greatly appreciated!