From a ZFC perspective, there is a unique set $\emptyset$, which is the empty subset of every set. Further, every set has a maximum subset, namely itself.

However from a structural perspective, there are many emptysets; indeed, each set $X$ has its own emptyset $\emptyset_X$, defined as the least element of the powerset of $X$. Of course each set $X$ also its own "full set" (or whatever this is called), which is the greatest element of the powerset of $X$. This is usually just denoted $X$, but I'd like to start writing it in the form $\top_X$ for some symbol $\top$, to emphasize that it is actually a different entity to $X$ in the conceptual system that I prefer.

Now in my opinion, the order-theorist's $\top_X$ isn't a good option for the maximum subset of $X$, and similarly $\bot_X$ isn't a good option for the empty subset of $X$. There are many reasons that I feel this way; for one, we should really be writing $\top_{\mathcal{P}(X)}$ as opposed to $\top_X$, but the former is too long-winded for my taste, and I don't like the way it forces us to explain powersets just to explain the empty subset and the maximum subset. Another reason I don't like $\top_X$ is that when we're doing lattice theory, readability is higher when we use symbols like $\emptyset,\cap,\cup$ for the relevant set-theoretic operations, while reserving the symbols $\bot,\top,\wedge,\vee$ for the operations in lattices other than powersets. (I think this is one of those rare situations where having more notation is actually less confusing.)

So anyway, I've started writing $\Omega_X$ for the maximum subset of $X$ in my own personal writing. However - and this is why I ask the question - if any authors have suggested notation for the maximum subset of $X$, I'd like to start using it instead of my $\Omega_X$. Good to promote worldwide notational consistency, you know? So:

Question. Has anyone proposed a symbol for the maximum subset of a set?

  • $\begingroup$ What sort of structural perspective do you have, where the various empty sets are not equal? $\endgroup$ – Tobias Kildetoft Jan 29 '15 at 9:36
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    $\begingroup$ Why are $\top$ and $\bot$ bad? $\endgroup$ – Git Gud Jan 29 '15 at 9:38
  • $\begingroup$ @TobiasKildetoft, the same perspective any topos theorist has. $\endgroup$ – goblin Jan 29 '15 at 9:45
  • $\begingroup$ @GitGud, I've added some further explanation of why I prefer not to use $\top$ and $\bot$ in this way. $\endgroup$ – goblin Jan 30 '15 at 14:16

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