Adding the example of unbounded monotone functions.

]]>Adding explicit definitions for preordered monoid and pomonoid. I hope this clears some of the confusion.

]]>https://ncatlab.org/nlab/show/pregroup+grammar

and it says:

If we drop the anti-symmetry axiom for posets, we get a quasi-pregroup,

but when I go to

https://ncatlab.org/nlab/show/partial+order

it says:

A poset is precisely a proset satisfying the extra condition that x≤y≤x implies that x=y.

Therefore, I think a quasi-pregroup is the same as a poset.

But this can't be right because the sub-words "group" and "set" don't mean the same thing.

I assume this is just a matter of being more precise, as tracking down

https://ncatlab.org/nlab/show/set

this set-page does NOT mention groups, but groupoids.

and

https://ncatlab.org/nlab/show/group

this group-page does not mention 0-groupoids, which I think are groups, right ?

but the group-page does mention a group is tied to a coset but not linked to the set-page

I'm wondering if I am drawing the wrong conclusions, or if some of these pages need a bit more duplication

to show when different combinations of main ideas and adjectives describing them cause you to just

get another name for an existing idea.

Thanks for the info,

Dave Whitten

713-870-3834

whitten@netcom.com ]]>

link to posets

DavidWhitten

]]>I fixed the omission of the homomorphic image part in my previous edit and added the respective references to Kobele and Kracht.

]]>Very likely my bad due to a false recollection of their argument, thanks for giving it a reality check! Hopefully, I got at least the substance right, namely, that product pregroup grammars land you in type 0. I’ll attend to the paragraph as soon as I find a minute. Feel free to revise concerning the Kracht-Kobele paper as you have a better overview of the relevant literature hence how the pieces fit together.

]]>Kobele, Gregory M., and Marcus Kracht. "On pregroups, freedom, and (virtual) conceptual necessity." University of Pennsylvania Working Papers in Linguistics 12.1 (2006): 16.

When you write "by a classical result in formal language theory any type 0 language is the intersection $L=L_1\cap L_2$ of two context-free languages $L_1,L_2$", are you referring to Theorem 1 in Kobele & Kracht? If so, the statement is a bit more subtle: any r.e. language is the image of an intersection of CFLs under a non-length-increasing homomorphism $L = h(L_1 \cap L_2)$.

I feel like that theorem is cool enough to deserve its own section somewhere, maybe in a new page on language homomorphisms? ]]>

I expanded on product grammars and attributed some results here to Kobele-Kracht. Since I am unfamiliar with the Lambek and Genkin et al. source my way to weave them in is somewhat conjectural: feel free to correct me on this!

]]>Adding a mention of mildly context-sensitive pregroup grammars.

]]>Add linguistics context.

]]>fix typos

Richie

]]>As far as I'm aware there isn't any reference which spells out the details fully, which is part of the reason why I started this page, as well as the one on context-free grammars. ]]>

Re #6: This blog post is saying something very similar, using a monoidal category for sentence derivations. “That is, a string $u \in V^\star$ is *grammatical* whenever there exists an arrow from the start symbol $s$ to $u$ in $\mathcal{C}_R$.” Morphisms in monoidal categories can be notated using string diagrams.

It should be pointed out (somewhere, sometime) the relationship between pregroups and residuated lattices, which appears in much older work of Lambek dating back at least to the 60’s. This work of Lambek was all of a piece, connecting module theory to proof theory to linguistics. I would like to add some of this history at some point.

]]>removing capital letter

]]>I'm starting to write down some notes, trying to connect them with what was already there on categorial grammars and linguistics in general. ]]>

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