What is the appropriate way to describe an "ordered union" of sequences?

Question

Given the $\langle $sequences$\rangle$: $$ \begin{aligned} S_1 & = \langle \color{magenta}{D_1}, \color{orange}{D_2}, D_3 \rangle \\ S_2 & = \langle \color{orange}{D_2}, D_4 \rangle \\ S_3 & = \langle \color{magenta}{D_1}, D_5, \color{dodgerblue}{D_1} \rangle \end{aligned} $$

1. Is there a standard way of referring to the sequence ${C_D}$ constructed by recording the elements of $S_1$, $S_2$ and $S_3$ in the order of their first appearance (including multiplicity) i.e.:

$$ C_{D} = \langle \color{magenta}{D_1},\, \color{orange}{D_2},\, D_3,\, D_4,\, D_5,\, \color{dodgerblue}{D_1} \rangle $$

2. Is there a standard way of describing the construction of $C_{D}$ from the initial subsequences? e.g. would it be correct to say:

$$ \underbrace{\langle \color{magenta}{D_1}, \color{orange}{D_2}, D_3 \rangle}_{S_1} \cup \underbrace{\langle \color{orange}{D_2}, D_4 \rangle}_{S_2} \cup \underbrace{\langle \color{magenta}{D_1}, D_5, \color{dodgerblue}{D_1} \rangle}_{S_3} = \langle \color{magenta}{D_1},\, \color{orange}{D_2},\, D_3,\, D_4,\, D_5,\, \color{dodgerblue}{D_1} \rangle = C_{D} $$

I've never heard of a "union of sequences" (and didn't find anything in a quick search), but I'm looking for the correct way of referring to the ordered union of the elements of $S_1$, $S_2$ and $S_3$, where new elements (or repeat elements of a higher multiplicity) are added to the end of the sequence, e.g.:

• E.g. 1: $$ S_1 \cup S_2 = \langle \color{magenta}{D_1}, \color{orange}{D_2}, D_3 \rangle \cup \langle \color{orange}{D_2}, D_4 \rangle = \langle \color{magenta}{D_1}, \color{orange}{D_2}, D_3, D_4 \rangle $$

  N.B.: $D_2$ is not repeated, because it occurs only once in each of $S_1$ and $S_2$.

• E.g. 2: $$ S_1 \cup S_3 = \langle \color{magenta}{D_1}, \color{orange}{D_2}, D_3 \rangle \cup \langle \color{magenta}{D_1}, D_5, \color{dodgerblue}{D_1} \rangle % = \langle \color{magenta}{D_1}, \color{orange}{D_2}, D_3, D_5, \color{dodgerblue}{D_1} \rangle $$ N.B.: $\color{dodgerblue}{D_1}$ of $S_3$ is added after $D_5$, because it is the second occurrence of $D_1$ in $S_3$ (i.e. its multiplicity is higher than the multiplicity of $\color{magenta}{D_1}$ in $S_1$); as explained in the context, this is analogous to recording factors of a common denominator with an imposed order.

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Context

I'm looking for the correct way to refer to the ordered factors of the common denominator of something like:

$$ \frac{N_1}{\color{magenta}{D_1} \cdot \color{orange}{D_2}} \cdot \frac{N_2}{{D_3}} + \frac{N_3}{\color{orange}{D_2} \cdot {D_4}} + \frac{N_4}{\color{magenta}{D_1} \cdot D_5 \cdot \color{dodgerblue}{D_1} }. $$

The common denominator of the terms is the product of the elements of the $[$multiset$]$:

$$ \big[ \color{magenta}{D_1},\, \color{dodgerblue}{D_1},\, \color{orange}{D_2},\, D_3,\, D_4,\, D_5 \big] $$

However, I'm looking for the correct way to describe the ordered sequence of the factors:

$$ C_{D} = \langle \color{magenta}{D_1},\, \color{orange}{D_2},\, D_3,\, D_4,\, D_5,\, \color{dodgerblue}{D_1} \rangle $$

as well as the correct term(s) for the operation(s) used in its construction.

Answer 1

(1) I think it's reasonably standard to call the larger sequence the "concatenation" of the smaller sequences. It's certainly an accurate term to use.

(2) I wouldn't use the union symbol, since that has a well-established different meaning for sets. I'd probably try to avoid a symbol at all and just describe it in words.

Answer 2

1. Is there a standard way of referring to the sequence constructed by recording the elements of 1, 2 and 3 in the order of their first appearance (including multiplicity) ... ?

I found a link to a part of Reversibility in Natural Language Processing by Gertjan van Noord that discusses the idea of a union of sequences:

[71], and [72] discuss a relation called sequence union to analyze discontinuous constituents. The sequence union of the sequences s1, s2 and s3 is true, iff each of the elements in s1 and s2 occur in s3, and moreover, the original order of the elements in s1 and s2 is preserved in s3. For example, the sequence union of the sequences $ \langle$a, b$ \rangle$ and $ \langle$c, d$ \rangle$ and s3 is true, iff s3 is any of the sequences: $$ \langle a,b,c,d\rangle\\ \langle a,c,b,d\rangle\\ \langle c,a,d,b\rangle\\ \langle c,a,b,d\rangle $$

However, this is broader than the operation described in the OP and the example provided does not make it clear how repeat elements ought to be treated.