Perhaps (for Joseph's definition of "somehow analogous") this could be a "Yes" answer. For those of us still struggling with the concepts, could you confirm/refute the assertion that "the classical case considered by Cayley" as you put it is the "second hyperdeterminant" as so-named in the Wikipedia article? Gerhard "Still Learning Geometry Of Hypermatrices" Paseman, 2015.10.08

Indeed. He reworked his research several times, with posts in 1890 and 1906. A copy of one of them can be found here: archive.org/details/theoryofdetermin01muiruoft . People were saying to him "Inter-net? Whereof speakest thou?", to which Muir calmly replied "Just wait." Gerhard "Clearly Ahead Of His Time" Paseman, 2015.10.08

In related questions on MathOverflow, including one Joseph asked some years ago, hypermatrices are (my interpretation) notational devices for tensors, and the formula for multiplication is derived and understood by students of hypermatrices, and bears some relation to composition of transformations. I think your statement about its having columns that are linearly independent in R^3 is wrong (I have a 0-1 example in mind), but I can't be sure since we haven't agreed on what it is you mean. Gerhard "Again, What Is A Column?" Paseman, 2015.10.07

I'm afraid slice is also not useful to me. For n=3, let there be a hypermatrix of 3x3x3=27 entries. I pick 3 of them at a time to embed in three space. I partition the hypermatrix entries into 9 such groups which I call "vectors". What do I do with these 9 vectors in 3-space that allows me to talk about linear independence? Take the vectors 3 at a time? Gerhard "Really, What Is A Slice?" Paseman, 2015.10.07

I'm sorry: what are you saying? I have trouble coming up with a notion of column that would embed in 3-space. Even if you embed it in n-space, I don't see that the n^2 many columns are linearly independent for n > 1. Gerhard "Please, What Is A Column?" Paseman, 2015.10.07

From the first paragraph on the Wikipedia article for hyperdeterminant: "Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes." I read that as a "No", but you may be looking for something more. Gerhard "It Is Wikipedia, After All" Paseman, 2015.10.07

Umm, Wikipedia has an entry (which on my reading says no to your question), and there have been some ArXiv articles computing hyperdeterminants for small cases which might also help with your question. Perhaps you mean to ask something else? Gerhard "Question Not Ready For Primetime?" Paseman, 2015.10.07

I think the Wikipedia entry of Cramer's rule is quite accessible and well motivated (and relevant here). Also, Thomas Muir posted a history of determinants (which existed before matrix notation!) and their development. You can find both resources on the web. Also, the question could use an example of what shape is wanted for the answer: pointers to the literature, a copy of the Wikipedia article, a category theory or foundational approach, or something understood by someone with only one or two linear algebra courses behind them. Gerhard "Falls Into The Last Category" Paseman, 2015.10.07

You might be able to prove existence by considering the associated lattices IDL(L) and FIL(L) for a given infinite distributive lattice L. I would not be surprised if this dealt with choice issues, e.g. that in a variant of set theory without choice one finds a model of this theory which believes it has an infinite distributive lattice all of whose prime ideals are principal. Gerhard "Still Capable Of Showing Surprise" Paseman, 2015.10.06

What happens if, during the running of this algorithm, the "none is a substring of another" condition is violated by $s_{i1},...,s_{ik}, b_{n-k+1}$, where the $b$ string is the newly formed big string? Or is that the point? Gerhard "Sometimes Not Quick On Uptake" Paseman, 2015.10.06

If the c_i's are not ordered, then you might find multinomial more useful than partition in searching. I would still look at partition as some sources might talk about the partition case being easier/different than the composition case (and give you something on compositions). Also, I read your post and (despite your worthy attempt) I still think partition. For the more dense among us, you might make it amazingly clear that Euler's partition function is not part of your picture. Gerhard "Not Quite 'MathOverflow For Dummies'" Paseman, 2015.10.05

I think you might find "integer partition" a more useful search term than "integer composition". (I am guessing that $c_i \leq c_j$ iff $i \leq j$.) You might find "multinomial" in combination with some other word also useful. Gerhard "Don't Know About The Sequence" Paseman, 2015.10.05

For that matter, there is no surjective order-preserving map from the completion of the naturals N to N as a lattice. It should be clear that unbounded and countably (maybe uncountably?) cofinal lattices cannot have such maps from their completions. Gerhard "Where Does The Top Go?" Paseman, 2015.10.01

$c(\sqrt{n})^n$ should be more like $c(\sqrt{n}/2)^n$, as I confused values for determinants of 0-1 matrices with values of their 1,-1 binary counterparts. In any case, I would like seeing even $(\log n)^n$ achieved with a uniform description. At present, I don't know how to make a short and uniform description do better than exponential growth. Gerhard "Just A Difference Of Two" Paseman, 2015.09.30

I'm noticing a large matricial aspect to the proposals. Gerhard "Glad I Proposed Mine Early" Paseman, 2015.09.30

Granted, I've moved away from it for now, but I may make it mine again, violating number (3). Sorry, Gil. Gerhard "Is Willing To Share Project" Paseman, 2015.09.30