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I don’t actually know the correct definition, but my spider sense tells me that in
“2. Isotone: If $F \supseteq G$ and $F \rightsquigarrow x$, then $G \rightsquigarrow x$;”
the $\supseteq$ should be $\subseteq$.
– Keith again
It's hard to converge to a point but easy to cluster to it. One way to phrase isotony for convergence is that a filter $F$ converges to a point $x$ iff every refinement of $F$ converges to $x$. Now, by definition (in a convergence space), $F$ clusters to $x$ iff some proper refinement of $F$ converges to $x$; but because refinement is transitive, it follows that $F$ clusters to $x$ iff some refinement of $F$ clusters to $x$. So that's reverse isotony for clustering.
For a specific example, let $F$ be the filter generated by the tails of the constant sequence $(0,0,0,0,\ldots)$ and let $G$ be the filter generated by the tails of the sequence $(0,1,0,1,\ldots)$. (Specifically, $A \in F$ iff $0 \in A$, and $A \in G$ iff $0, 1 \in A$.) We have $G \subseteq F \to 0$ but not $G \to 0$, so isotony can't run that way for convergence. But we have $F \supseteq G \rightsquigarrow 1$ but not $F \rightsquigarrow 1$, so isotony can't run that way for clustering.
Hi Keith!
There are two meanings of ‘isotone’ in the literature (as far as I've seen):
Either way, ‘antitone’ means the opposite (montone decreasing, or with a counit). This gives ‘antitone’ two meanings as well, although I think that this word gets more usage (mostly in its first meaning).
As a synonym of ‘monotone’, it's not entirely useless, since people sometimes use ‘monotone’ to meaning either increasing or decreasing, while ‘isotone’ only means increasing. (Compare why ‘monotone’ itself is not a useless synonym of ‘increasing’: sometimes people use that to mean strictly increasing.) With the other meaning, it's not useless at all; unfortunately, the two meanings clash. (They are related; given $f\colon X \to X$, you get a map from $\mathbb{N}$ to $X \to X$ that takes $n$ to $f^n$; this map is monotone increasing iff $f$ has a unit.) You can see both ‘monotone’ and ‘isotone’ (with the second meaning) used together at Moore closure#InTermsOfClosureOperators.
In this case, the meaning is the first, so why not say ‘monotone’? My only answer is that the literature on convergence spaces says ‘isotone’ (at least as far as I've seen^{1}). It seems particularly perverse to use it in the axioms for clustering spaces, since this is actually decreasing, so we should say ‘antitone’. (Muscat, the original reference on cluster spaces, does not use any term for this; it was introduced by whoever made the anonymous edit putting this subject on the Lab). And since we're thinking of these more as relations between $\mathcal{F}X$ and $X$ rather than as maps from $\mathcal{F}X$ to $\mathcal{P}X$, ‘upper’ and ‘lower’ might be even better, as you said.
And all of this terminology can be especially problematic here, because people don't always agree on the natural order on the set of filters! Subset inclusion is the most obvious, but there is a case to be made that refinement is more appropriate, and this is containment, the reverse of inclusion. I believe that I've even seen literature that defined $F \leq G$ (for filters $F$ and $G$) to mean $F \supseteq G$ (although I might be thinking of something else). That's actually a case for using ‘monotone’, because it allows for this ambiguity.
For now, I'll change ‘isotone’ to ‘antitone’ in the definition of cluster spaces. You can make the case that it's isotone, because filters should be ordered by refinement, but then you shouldn't use that word for convergence spaces as well. But I could easily see using ‘monotone’ for both of them, especially if we can find some literature on convergence spaces that already does this. I think that ‘lower’ and ‘upper’ would be more confusing than ‘monotone’ without context, so I really wouldn't want to use them unless, again, we have some literature on convergence spaces that already uses ‘upper’ in this way.
Or are you citing Johnstone and Gierz as using ‘upper’ and ‘lower’ in a discussion of convergence spaces? If you're simply citing them for ‘upper set’ and ‘lower set’, then I agree, that's standard, and we use those terms on the Lab. But if they use them as a term for this condition in a convergence space, then that would be even better. Even the intermediate case, as terminology for a property of a relation between a p(r)oset and a set, would be good. ↩
Hi,
for a map between any two p(r)osets, a synonym of ‘monotone’
I guess that p(r)oset means either poset or proset, and that proset means pre-ordered set. I definitely approve of that term; I thought I invented it myself after reading some category theorist who wanted to re-define “poset” on the grounds that it is un-categorical to force iso to be equal. I never published, or even told anyone, so it’s good to see that others heve thought of it too.
On the other hand, why do we need a synonym of monotone? Every child knows what monotone means. Say “increasing” once for clarity, and then drop it.
for a map from a p(r)oset to itself, the condition that
the map has a unit: $x \leq f(x)$ always.
I would call such a map ’inflationary’. I think I read that somewhere, but I don’t have a reference handy.
But what do either of those meanings have to do with axiom 2?
Or are you citing Johnstone and Gierz as using ‘upper’
and ‘lower’ in a discussion of convergence spaces?
If you're simply citing them for ‘upper set’ and ‘lower set’,
The later. Specifically, let $S_x$ be the set of filters that converge to $x$, $S_x = \{F | F \to x \}$. For every $x$, $S_x$ is an upper set in the poset of filters ordered by inclusion (or refinement). That is exactly part 2. of the definition.
Where is the monotone function?
Subset inclusion is the most obvious, but there is a case
to be made that refinement is more appropriate, and this
is containment, the reverse of inclusion.
Why do you say that? For both a topology (set of open sets) and a filter (subset of $P(X)$) the finer includes the coarser as a subset[1]. Do you have some reason to want to say fine $\le$ coarse? I think more refined is greater.
[1] Steen \& Seebach, Counterexamples in Topology
But what do either of those meanings have to do with axiom 2?
Where is the monotone function?
The convergence relation defines a map from $\mathcal{F}X$ (the sets of filters on $X$) to the $\mathcal{P}X$ (the power set of $X$), and this is the map that (2) says is monotone.
Do you have some reason to want to say fine $\leq$ coarse?
In ordinary language, something fine is smaller than something coarse. For fine and coarse filters, it's really that the fine filter is made of smaller sets than the coarse filter. But that's what this convention is meant to achieve: the lattice of filters behaves like the sets inside the filters. For example, the improper filter is the bottom element, rather than the top, and this matches that it owns the empty set, which is the bottom subset; in other words, a proper filter, one in which each element is positive (inhabited, nonempty), is itself positive (not the bottom) in the lattice of filters. Conversely, the top filter is the one that owns only the top subset. Similarly, the meet of $F$ and $G$ is $\{A \cap B \;|\; A \in F,\; B \in G\}$, while the join is generated by $\{A \cup B \;|\; A \in F,\; B \in G\}$. (Thus, both of these have been written as $F \vee G$; and both of them have also been written as $F \cap G$. Having different conventions can certainly be confusing!)
I think that it's even nicer for filterbases. If $F$ and $G$ are filterbases, and $F$ is a refinement of $G$, then it's not necessarily true that $F \supseteq G$, but it is true that for each $A$ in $F$, for some $B$ in $G$, $A \subseteq B$, and this is reflected in writing $F \leq G$. $F$ is positive (not a bottom, proper) iff for each $A$ in $F$, $A$ is positive (inhabited, not empty); $F$ is a bottom (improper) iff for some $A$ in $F$, $A$ is empty. The meet of $F$ and $G$ is still $\{A \cap B \;|\; A \in F,\; B \in G\}$, and now the join of $F$ and $G$ is precisely $\{A \cup B \;|\; A \in F,\; B \in G\}$. (Of course, filterbases form a proset rather than a poset, so meets and joins are only defined up to equivalence, but these operations give one a canonical meet and join that work.)
OK, that makes sense. I’m not sure I will adopt the terminology, but I am done discussing it for a while.
I came to this web page while reading a paper [1] which defines a filter space in terms of an operation which associates, to each point, a collection of filters. This page uses the equivalent formulation in terms of a relation between filters and points.
But a relation $R \subseteq A \times B$ can be seen as a function in two ways: $R^{+1}: \mathcal{P}A \leftarrow B$ and $R^{-1}: A \rightarrow \mathcal{P}B$. If $A$ is a p(r)oset then the range of $R^{+1}$ comprises only upper sets iff $R^{-1}$ is monotone. It seems that I was thinking of one of those functions while you were thinking of the other.
I wonder if it would help to be more idealistic (instead of filteristic) or whether we are doomed by the extended Yoneda Lemma:
A category embeds covariantly into a contravariant functor category, and contravariantly into a covariant functor category, and furthermore, no matter how you bang on it to try to straighten it out, it will be contra-something.
[1] Hyland, Filter Spaces and Continuous Functionals, Ann. Math. Logic 16 (1979)
— Keith Wright
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