<p>Papers I've seen that have this issue often present 1 as the initial definition and then also define 2 (or 3 with <em>subscripts</em>) as an additional definition that is used when necessary for clarity.</p>

<p>A common abuse of notation—common enough that I've seen it in multiple places, although it makes my skin crawl every time I see it—is to write something like <span class="math-container">$S = S(TS)$</span>.</p>

<p>3 with <em>superscripts</em> is unacceptable because superscripts have other widely-understood standard meanings; however, it would be fine if the superscripts were changed to subscripts. 4 is never used outside of programming contexts.</p>

<p>I would also recommend avoiding the two-letter variable name <span class="math-container">$TS$</span> in mathematical notation, especially since you are also using <span class="math-container">$S$</span> as a separate symbol. Use single letters, resorting to Greek if necessary, for anything in the standard italic "variable" font. Special functions and classes of languages usually get straight font or boldface to distinguish them from surrounding text for this reason.</p>

CC BY-SA 4.0
2019-12-27T03:28:13.183Z
I have seen style 4 used in cryptography , for example when the object at hand is a signature scheme or similar. See page 13 of https://www.iacr.org/archive/tcc2016b/99850156/99850156.pdf as an example.
Robbo
CC BY-SA 4.0
2019-12-23T22:15:18.873Z
2019-12-23T22:15:18.873Z
Aaron Rotenberg
<p>I have the need to define a multi-component object type for which I provide numerous definitions throughout the text.
However, in most of these definitions, I only need to refer to one or two of the components constituting the object type.
Therefore, I am looking for a widely used notation that allows to quickly refer to only a single component of the object type without having to introduce the entire object.</p>

<p>I have identified four possible notations for defining multi-component objects. which I do introduce below using the object type <em>transition system</em> as an example.
The first notation is the classical one where the object type is defined as a tuple. The drawback of this definition is that anytime that I want to express properties on the object type I need to re-introduce the entire tuple.
The other three notations (2-4) avoid this issue, but I am not sure whether they are widely used and accepted.</p>

<p>Here are a few options</p>

<ol>
<li><blockquote>
  <p>A transition system <span class="math-container">$TS$</span> is a tuple <span class="math-container">$(S,\Sigma,\delta, s_0)$</span> where</p>
  
  <ul>
  <li><span class="math-container">$S$</span> is a set of states</li>
  <li><span class="math-container">$\Sigma$</span> is a set of Actions</li>
  <li><span class="math-container">$\delta \subseteq S \times \Sigma \times S$</span> is the state transition relation</li>
  <li><span class="math-container">$s_0 \in S$</span> is the initial state</li>
  </ul>
</blockquote>

<p><strong>Usage example</strong></p>

<blockquote>
  <p>A transition systems <span class="math-container">$TS = (S,\Sigma,\delta, q_0)$</span> is finite if <span class="math-container">$S$</span> is finite</p>
</blockquote></li>
<li><blockquote>
  <p>A transition system <span class="math-container">$TS$</span> is consists of</p>
  
  <ul>
  <li>a set of states <span class="math-container">$S(TS)$</span></li>
  <li>a set of actions <span class="math-container">$\Sigma(TS)$</span> </li>
  <li>a transittion relation <span class="math-container">$\delta(TS) \subseteq S(TS) \times \Sigma(TS) \times S(TS)$</span> </li>
  <li>the initial state <span class="math-container">$s_0(TS) \in S(TS)$</span> </li>
  </ul>
</blockquote>

<p><strong>Usage example</strong></p>

<blockquote>
  <p>A transition systems <span class="math-container">$TS$</span> is finite if <span class="math-container">$S(TS)$</span> is finite</p>
</blockquote></li>
<li><blockquote>
  <p>A transition system <span class="math-container">$TS$</span> is consists of</p>
  
  <ul>
  <li>a set of state <span class="math-container">$S^{TS}$</span> </li>
  <li>a set of action <span class="math-container">$\Sigma^{TS}$</span> </li>
  <li>the transition relation <span class="math-container">$\delta^{TS} \subseteq S^{TS} \times \Sigma^{TS} \times S^{TS}$</span> </li>
  <li>the initial state <span class="math-container">$s_0^{TS} \in S^{TS}$</span> </li>
  </ul>
</blockquote>

<p><strong>Usage example</strong></p>

<blockquote>
  <p>A transition systems <span class="math-container">$TS$</span> is finite if <span class="math-container">$S^{TS}$</span> is finite</p>
</blockquote></li>
<li><blockquote>
  <p>A transition system <span class="math-container">$TS$</span> is consists of</p>
  
  <ul>
  <li>a set of states <span class="math-container">$TS.S$</span></li>
  <li>a set of actions <span class="math-container">$TS.\Sigma$</span> </li>
  <li>a transittion relation <span class="math-container">$TS.\delta \subseteq TS.S \times TS.\Sigma \times TS.S$</span> </li>
  <li>the initial state <span class="math-container">$TS.s_0 \in TS.S$</span> </li>
  </ul>
</blockquote>

<p><strong>Usage example</strong></p>

<blockquote>
  <p>A transition systems <span class="math-container">$TS$</span> is finite if <span class="math-container">$TS.S$</span> is finite</p>
</blockquote></li>
</ol>

<notation>
What are the widely used notations for quickly referring to a single component of a multi-component object?
Robbo
# What are the widely used notations for quickly referring to a single component of a multi-component object?

(Asked by: username_0 on 2019-12-22)

I have the need to define a multi-component object type for which I provide numerous definitions throughout the text.
However, in most of these definitions, I only need to refer to one or two of the components constituting the object type.
Therefore, I am looking for a widely used notation that allows to quickly refer to only a single component of the object type without having to introduce the entire object.

I have identified four possible notations for defining multi-component objects. which I do introduce below using the object type *transition system* as an example.
The first notation is the classical one where the object type is defined as a tuple. The drawback of this definition is that anytime that I want to express properties on the object type I need to re-introduce the entire tuple.
The other three notations (2-4) avoid this issue, but I am not sure whether they are widely used and accepted.

Here are a few options

1. > A transition system $TS$ is a tuple $(S,\Sigma,\delta, s\_0)$ where
   >
   > * $S$ is a set of states
   > * $\Sigma$ is a set of Actions
   > * $\delta \subseteq S \times \Sigma \times S$ is the state transition relation
   > * $s\_0 \in S$ is the initial state

   **Usage example**

   > A transition systems $TS = (S,\Sigma,\delta, q\_0)$ is finite if $S$ is finite
2. > A transition system $TS$ is consists of
   >
   > * a set of states $S(TS)$
   > * a set of actions $\Sigma(TS)$
   > * a transittion relation $\delta(TS) \subseteq S(TS) \times \Sigma(TS) \times S(TS)$
   > * the initial state $s\_0(TS) \in S(TS)$

   **Usage example**

   > A transition systems $TS$ is finite if $S(TS)$ is finite
3. > A transition system $TS$ is consists of
   >
   > * a set of state $S^{TS}$
   > * a set of action $\Sigma^{TS}$
   > * the transition relation $\delta^{TS} \subseteq S^{TS} \times \Sigma^{TS} \times S^{TS}$
   > * the initial state $s\_0^{TS} \in S^{TS}$

   **Usage example**

   > A transition systems $TS$ is finite if $S^{TS}$ is finite
4. > A transition system $TS$ is consists of
   >
   > * a set of states $TS.S$
   > * a set of actions $TS.\Sigma$
   > * a transittion relation $TS.\delta \subseteq TS.S \times TS.\Sigma \times TS.S$
   > * the initial state $TS.s\_0 \in TS.S$

   **Usage example**

   > A transition systems $TS$ is finite if $TS.S$ is finite

**Tags:** notation


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## 1 answers

### Answer by username_1 on 2019-12-23. Score: 4

Papers I've seen that have this issue often present 1 as the initial definition and then also define 2 (or 3 with *subscripts*) as an additional definition that is used when necessary for clarity.

A common abuse of notation—common enough that I've seen it in multiple places, although it makes my skin crawl every time I see it—is to write something like $S = S(TS)$.

3 with *superscripts* is unacceptable because superscripts have other widely-understood standard meanings; however, it would be fine if the superscripts were changed to subscripts. 4 is never used outside of programming contexts.

I would also recommend avoiding the two-letter variable name $TS$ in mathematical notation, especially since you are also using $S$ as a separate symbol. Use single letters, resorting to Greek if necessary, for anything in the standard italic "variable" font. Special functions and classes of languages usually get straight font or boldface to distinguish them from surrounding text for this reason.

> **Comments:**
> **username_0**: I have seen style 4 used in cryptography , for example when the object at hand is a signature scheme or similar. See page 13 of https://www.iacr.org/archive/tcc2016b/99850156/99850156.pdf as an example.  

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