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I found and added a reference to a Trans AMS paper from 1939 on residuated lattices. That paper mentions some work of Krull, but I have not got the precise reference nor know what Krull did in this area.
Just to say that if the motivation is to follow up on the issue raised in the thread (here) on the broken entry “residual”, then this entry here is still lacking a remark that its first displayed math line defines an internal hom in the lattice regarded as a symmetric monoidal $(0,1)$-category. Let me know if you plan to do it. Otherwise, I’ll put it on my to-do list.
The story of Residuated Lattices starts before Category Theory, even though residuals are adjoints. The prime example of a residuated mapping in the ceiling mapping. Residual suggests ‘left over’ so interpretation as an internal hom is really very neat. I will see what I can dream up for the wording. Perhaps some work on residual is also needed. The current form of the idea section is too specialised without quite a lot of changes.
It seems to me that all this is a very neat example of that process by which category theory unifies apparently unrelated ideas by means of some form of categorification.
Fixed some glitches and typos.
I have written something on the categorical interpretation of residuated lattices, but have not gone as far as you suggested as I am wondering if the point you made might not be better in residual.
I intend to write a short entry on lattice ordered groups and link that to here. The residuals then are the obvious quotients, if I remember rightly, but must pause for the moment.
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