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The previous discussion was too lengthy. So I start a new thread.
Based on above mentioned Todd Trimbles comments, I have proved (see the draft of volume 2 of my book, chapter 4) that the set of prestaroids between join-semilattices (that is maps $\prod_{i\in n} S_i\rightarrow 2$ preserving joins in separate arguments) is order-isomorphic to a set of ideals on a poset.
If you have a little free time, you may check this my proof for errors. (However, it seems OK.)
Now I define more general prestaroids between posets:
Free stars on a poset $S$ are sets $F\in\mathcal{P}S$ such that to $F\ne\mathcal{P}S$ and $A,B\in\overline{F}\Leftrightarrow\exists Z\in\overline{F}:(Z\ge A\wedge Z\ge B)$.
Equivalently, free stars are sets $F\in\mathcal{P}S$ such that $F\neq\mathcal{P}S$ and $\forall Z\in S:(Z\ge A\wedge Z\ge B\Rightarrow Z\in F)\Leftrightarrow A\in F\vee B\in F$ for every $A,B\in S$.
For arbitrary (possibly infinite) index set $n$ a prestaroid of arity $n$ is a subset of $\prod_{i\in n} S_i$ such that, if we fix all but one argument, the allowed values for the non-fixed argument is a free star.
Conjecture The set of prestaroids between any posets is isomorphic to the sets of ideals on a poset.
Dear Victor Porton,
it seems quite unlikely to me that you will find here, or elsewhere, a correspondent that could be more valuable to your cause than Todd Trimble. The feedback that he provided in the thread that you seem to be abandoning hereby was of exceptional quality, especially regarding the subject and the methods that you are interested in. The insight that Todd had provided in that thread was very worthwhile all in itself, and it seems hard to imagine that you could hope to get anything close to that by restarting to fish for comments here. I don’t see how it could be wise for you not to try what you can to continue the conversation with Todd. Maybe it’s too late anyway, of course, but since you are opening up this new thread here for what seems to me to be no good reason, I thought I’d make that comment.
Also, might I add a general comment about research: you asked a question, and you got an interesting answer. It turned out that you had a slightly different question in mind, so you ignored the answer. Successful and interesting research doesn’t ignore such leads. You had a chance to a) learn something that would be useful (the theory of symmetric monoidal closed categories, and in particular one related to your work) and b) find a connection between your work and something other people are interested in, and you just dropped Todd after he spent the effort writing what would be an excellent seminar or two at the graduate student level.
I, like Urs, doubt that someone else will jump in and answer your questions given how you responded to Todd’s help. Please reconsider your use of the nforum as a fishing ground for help to prove results: it is your research after all, not ours.
I wish you all the best with finding non-trivial examples for your theory. I mean this sincerely: it’s taken me ten years, on and off, to get to a point where I can find the constructions I wanted to do for my PhD thesis.
David,
I’ve formulated my question more or less correctly: I have asked whether my $Strd(-,-)$ is a tensor product of posets (replacement of my question with whether it is a tensor product in the category of semilattices would be also good). I have not asked whether product of semilattices exists.
something that would be useful
I’ve already said that reading Todd’s responses was a good exercise for me. I am really thankful to Todd.
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