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What is the meaning of $\sqsubset$ and $\sqsubseteq$? Does it have any special meaning, or is it just an alternative to writing $\subset$ and $\subseteq$ respectively (for proper subsets and subsets)?

I have been looking for an explanation everywhere, but so far I could not find it. This may have to do with the fact that I am not even sure what this symbol is called (makes it difficult to search for it), but I have tried several things (like searching for $\sqsubset$ on this site), and nothing came up, other than lists of mathematical symbols for LaTeX without any explanation.

I have seen it used in papers (e.g., http://www.cril.univ-artois.fr/~marquis/everaere-konieczny-marquis-ecai10.pdf on page 4, footnote 5), but never explained. I am starting to think that $\sqsubset$ and $\sqsubseteq$ are equivalent to $\subset$ and $\subseteq$. However, sometimes there are subtle differences, so I want to be certain about this. I want to be sure that I understand the intended meaning when reading future papers, to avoid any misunderstandings.

Thanks in advance.

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  • $\begingroup$ The paper you linked seems to define the symbol as inclusion of profiles. The definition is in the Preliminaries section at the end of the fourth paragraph. $\endgroup$
    – Jorik
    Dec 10, 2015 at 15:55
  • $\begingroup$ From the context, it is clear that this is simply the subset symbol- possibly in an unfortunate typeface! $\endgroup$
    – user247327
    Dec 10, 2015 at 15:57
  • $\begingroup$ No it seems to be a little stronger. Since the order of the $K$ and $K'$ elements matters. $\endgroup$
    – Jorik
    Dec 10, 2015 at 16:00
  • $\begingroup$ So, if I understand it correctly, $E \sqsubseteq E'$ iff $E = E'$, or $E$ is contained in $E'$ such that the first element to some element in $E'$ is equal to $E$ (without leaving any "gaps"). It is also about vectors rather than sets, as I understand it. $\endgroup$
    – Jochem
    Dec 10, 2015 at 16:18
  • $\begingroup$ @Jochem I haven't read the entire paper but it seems to be something along those lines. But at least you have a definition now. $\endgroup$
    – Jorik
    Dec 10, 2015 at 16:21

2 Answers 2

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As far as I know, there is no universally accepted meaning for $\sqsubset$ or $\sqsubseteq$. If you see it in a book or article, it will have to be defined by the author in-context.

In general, there are a ton of symbols available in LaTeX (e.g. $\precsim$, $\oplus$, $\curlyvee$) that don't have well-agreed-upon meanings. These are there so that authors have access to plenty of characters to define their own operators.

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    $\begingroup$ $\sqsubset$, $\sqsubseteq$, sqsuperset, and sqsuperseteq (the "reverse" relations) are labeled square image of, square image of or equal to, square original of, and square original of or equal to respectively. Finding accessible definitions for these terms via search engine is proving difficult and likely for good reason! $\endgroup$
    – Corellian
    Dec 10, 2015 at 17:04
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The square subset symbol is sometimes used to indicate a prefix, so that $x \sqsubseteq y$ denotes that $x$ is a prefix of $y$. This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

This interpretation seems to make sense for the example you cited:

$(E_n)_{n \in \mathbb{N}}$ satisfies $\forall i \in \mathbb{N}, E_i \sqsubseteq E_{i+1}$

meaning $E_i$ is a prefix of $E_{i+1}$.

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  • $\begingroup$ This is also consistent with the usage of ⊏ or ⊑ in "Introduction to Algorithms" by Cormen, Leiserson, Revist, and Stein. $\endgroup$
    – Justin Raymond
    Feb 2, 2020 at 14:18
  • $\begingroup$ Cormen et. al is also what brought me here. They use $\sqsubset$ and $\sqsupset$ to refer to string prefix and suffix relationships respectively. Ie "aab" $\sqsubset$ "aabcde" and "cde" $\sqsupset$ "aabcde" $\endgroup$
    – Cole
    May 23, 2021 at 23:14
  • $\begingroup$ See their definition on pg 986 of the third edition. $\endgroup$
    – Cole
    May 23, 2021 at 23:25

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