Does $\sqsubset$ have any special meaning?

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.

Solution 1:

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.

Solution 2:

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 $E_{i+1}$.