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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×00p:A\times 0\to 0], the unique arrow !:0A!:0\to A [sic]. Certainly we have p!=1p \circ ! = 1. On the other hand, the distributivity axiom establishes A×(0+0)A\times (0+0) as the coproduct of A×0A\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 !p=1!\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 5K_{5} and K 3,3K_{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 CC 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 CC. The best known special case might be Kruskal’s theorem, on the class of finite forests, where the triangle K 3K_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( iy i)= i(xy i) 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 aba\Rightarrow b (or exponential object b ab^a) is:

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

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

    x( iy i)= i(xy i) 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\v)= uvxx(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.

    (uv)= xuvx=(u\v)= uvxx(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\0=uu \backslash 0 = u, whereas u0u \Rightarrow 0 is the negation of uu. The variance is also off: uvu \Rightarrow v is contravariant in uu and covariant in vv; it’s the other way around for u\vu \backslash v. In the Boolean case, we actually have u0=1\uu \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.

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