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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeJun 8th 2015
• (edited Jun 8th 2015)

Spurred by an MO discussion, I added the observation that coproduct inclusions are monic in a distributive category.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJun 11th 2015

I added a bit more to distributive category, including a proof that distribution over binary coproducts implies distribution over nullary coproducts.

I don’t know what the standard proof of that is. The proof I wrote down is a little tricky.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeJun 11th 2015

I added a link to extensive category at the bottom, but it might be worth mentioning near the beginning.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJun 12th 2015

Here’s the proof from Carboni-Lack-Walters:

There is only one possible inverse [to $p:A\times 0\to 0$], the unique arrow $!:0\to A$ [sic]. Certainly we have $p \circ ! = 1$. On the other hand, the distributivity axiom establishes $A\times (0+0)$ as the coproduct of $A\times 0$ with itself, the coprojections being equal. But any sum with coprojections equal can have at most one arrow to any other object and so $!\circ p = 1$.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJun 12th 2015

Ah, okay, thanks for that Mike. That seems preferable.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeMay 3rd 2016

Curiously, distributive category and distributive lattice were not cross-linked, so I fixed that.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeOct 12th 2016
• (edited Oct 12th 2016)

Looking over some old Café conversation (round about here and following), I was moved to add some material on alternative (manifestly self-dual) criteria, leading up to Birkhoff’s forbidden sublattice criterion, to distributive lattice.

Nowhere could I find online arguments for some of this stuff, such as the cancellation criterion, that are cost-free, clear, and constructively valid. So putting this in is my good deed for the nLab this day. :-) There is more to be added to modular lattice as well.

In Indiscrete Thoughts, Rota tells a story about some illustrious mathematician who came up to him and shouted, “Admit it! All lattice theory is trivial!” Yeah, I really don’t think so. I’ve just come to learn in the past day or two of one problem in lattice theory that was open for more than 4 decades (1904-1945), due to E.V. Huntington; it asks whether a lattice with unique complementation has to be distributive. It sounds at first like it might not be so bad. But it’s a bear. And to this day no example in the wild is known (the answer to the question is negative, but Dilworth’s 1945 solution which came as a complete surprise is a grueling syntactic analysis, based on Whitman’s seminal work on free lattices, and even today the proof is pretty hard work).

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeOct 12th 2016

Neat; thanks!

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeOct 12th 2016

Do we have a page about “forbidden subobjects”? In addition to Birkhoff’s sublattices blocking distributivity, and the classical $K_{5}$ and $K_{3,3}$ blocking planarity of a graph, there is the “pinwheel configuration” that blocks composability of a brick diagram in a double category; and what others?

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeOct 12th 2016

I don’t know that we have such a page; it’s certainly worth considering. There is however a major theorem in graph theory, the graph minor theorem of Robertson and Seymour, that says that for any class $C$ of finite graphs that is closed under taking graph minors, there is a finite collection of “forbidden graphs” that cannot appear as a minor of any element of $C$. The best known special case might be Kruskal’s theorem, on the class of finite forests, where the triangle $K_3$ is the forbidden graph. We have a smidgen about this in the nLab, here.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeOct 13th 2016

Thanks for the pointer. I wonder whether the forbidden sublattices and pinwheels can be seen as cases of the forbidden minor theorem?

• CommentRowNumber12.
• CommentAuthorRodMcGuire
• CommentTimeOct 14th 2016

i added to distributive lattice the following section and modified the finite distributive section to note that such lattices are bi-Heyting. I probably made some minor errors or may be even wronger.

I haven’t modified or cross linked any of the related pages: Heyting algebra, co-Heyting algebra, frame, locale, completely distributive lattice.

It seems somewhat strange that Heyting algebra and co-Heyting algebra don’t mention infinite distribution or give formulas for the exponentials.

## Infinitely distributive property

A distributive lattice that is complete (not necessarily completely distributive) may be infinitely distributive or said to satisfiy the infinite distributive law :

$x \wedge (\bigvee_i y_i) = \bigvee_i (x\wedge y_i)$

This property is sufficient to give the lattice Heyting algebra stucture where the implication $a\Rightarrow b$ (or exponential object $b^a$) is:

$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x$

Note that this property does not imply the dual co-infinitely distributive property:

$x \vee (\bigwedge_i y_i) = \bigwedge_i (x\vee y_i)$

Instead this dual gives the lattice co-Heyting structure where the co-implication or “subtraction” ($\backslash$) is

$(u \backslash v) = \bigwedge_{u \leq v \vee x} x$

If a lattice has both properties, as in a completely distributive lattice, then it has bi-Heyting structure (both Heyting and co-Heyting) and the two exponentials are equal.

$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x \qquad = \qquad (u \backslash v) = \bigwedge_{u \leq v \vee x} x$
• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeOct 14th 2016

For future reference, there is no need to copy on the forum the text added – anyone can follow the link and see it for themselves.

• CommentRowNumber14.
• CommentAuthorTodd_Trimble
• CommentTimeOct 14th 2016

Re #12: it looks fine, except that if it were me, I’d be more inclined to put this mostly at frame, with a quick mention and link from distributive lattice. Definitely the calculation of the exponential would improve the article frame, which at present contents itself with a quick glossing mention of adjoint functor theorem.

However, it doesn’t make sense to say those two exponentials agree, not even in the Boolean case. For example $u \backslash 0 = u$, whereas $u \Rightarrow 0$ is the negation of $u$. The variance is also off: $u \Rightarrow v$ is contravariant in $u$ and covariant in $v$; it’s the other way around for $u \backslash v$. In the Boolean case, we actually have $u \Rightarrow 0 = 1 \backslash u$.

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeOct 16th 2016

It seems somewhat strange that Heyting algebra and co-Heyting algebra don’t mention infinite distribution or give formulas for the exponentials.

Actually, Heyting algebra does both. Look carefully here.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

I have added (here) the (small) list of examples as grabbed form rig category.

But really some counter-examples should be mentioned…

• CommentRowNumber17.
• CommentAuthorThomas Holder
• CommentTimeJan 8th 2021

Added further examples and non-examples.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJan 8th 2021
• (edited Jan 8th 2021)

Thanks!!

Should we make a page for your example of “rectangular bands”? I had never heard of these before.

Just on formatting: Where you had

  $xx= x$ for all $x\in X$ and $xyx=x$


I have inserted whitespace

  $x x = x$ for all $x\in X$ and $x y x=x$


because otherwise Instiki renders the products as variable names typeset in mathrm

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeJan 8th 2021
• (edited Jan 8th 2021)

rectangular bands

They cropped up before as algebras for some version of the reader monad.

• CommentRowNumber20.
• CommentAuthorThomas Holder
• CommentTimeJan 8th 2021
• (edited Jan 8th 2021)

They are definitely worth an entry. A look at the old catlist-dicussion shows that they pop up as example for a lot of different things like affine theories, collapsed toposes, in Barr’s “point of the empty set”-paper, they also share the associativity-for-free-property with graphic monoids. For the terminology I don’t know, I used the name suggested by Peter Johnstone there and apparently taken from semigroup theory.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJan 8th 2021
• (edited Jan 8th 2021)

Okay, so I made rectangular band a hyperlink. Will create an annoyingly empty stub for it now, hoping to prompt you to add some content…

• CommentRowNumber22.
• CommentAuthorGuest
• CommentTimeJan 8th 2021
Yemon here (have forgotten, or perhaps never created, my password). I'm not sure if David's comment above is referring to bands or to the particular subspecies known as rectangular bands. A band is an idempotent semigroup, but a rectangular band is much more specialised. In between the two notions one has the concept of a left-regular band, whose semigroup algebras turn out to be related to random walks on certain hyperplane arrangements (not very categorical, I know)...
• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

Thanks, Yemon. As per the other thread (here), now both band and rectangular band point to the same entry, containing essentially the material you provided. Any further disambiguation can be written there. Or if the entries should be split, in two (or more) please do (or let me know if I should do it).

• CommentRowNumber24.
• CommentAuthorUlrik
• CommentTimeJan 9th 2021

There should probably also be some disambiguation to the band of a gerbe.

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeJan 9th 2021
• (edited Jan 9th 2021)

Re #22, oh, I see I didn’t put the link in properly in #19 to Sam Staton’s comment.

If I recall correctly, when $E=2$ then the algebras are precisely the rectangular bands.

1. Added a missing assumption. In order for an extensive category to be distributed it needs to have finite products.

Tomas Jakl

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