Merge operation in homological algebra?

Question

I provide you with a definition for the Merge operation in one standard textbook on the minimalist program in linguistics:

Merge: "basic structure-building mechanism. Merge takes two elements A and B and forms a two-membered set labeled C. C can then be merged with another element. Merge is said to be external if A and B are previously unconnected elements.. Merge is said to be internal if A combines with B and B contains D with which A had previously combined. Merge is said to be parallel if A merges with both B and D at a point in the derivation when B and D are not connected with one another.”

How would you formalize that in homological algebra?

PS: I should maybe add that the standard (but by no means only and / or necessary) representation of relations within the framework is given by binary trees (sometimes by labeled brackets as well, which are intended to translate exactly the same information but are more cumbersome to use extensively).

http://en.wikipedia.org/wiki/Merge_%28linguistics%29

Would you say thar description of Merge qualifies as a syzygy?

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I have founded this reference:

http://arxiv.org/abs/math/0507077

Any clue as to whether that might go in the right direction as to a mathematical treatment of Merge as presented above?

Answer 1

Disclaimer: not an answer.

Since I have no idea what you really are talking about, I may just play around with what I guess you are talking about using representation theory (which can be regarded as homological algebra in some sense.)

It will not be easy to explain the exact mathematics, but just watch the combinatorics evolve:

Your first definition (external merge) seems to say you want to glue two distinctive things together. This is like taking extension.

Let $Q$ be the quiver $1\to 2$, there are three indecomposable representations of the quiver $Q$ which can be drew in the following way: $1$, $2$ $\substack{1\\ 2}$. So the third representation (diagram) is like gluing $1$ and $2$ together. In homological algebra, this third representation (=module) is the extension of simple module corresponding to $1$ by the simple module corresponding to $2$; it also corresponds to the unique basis element of $Ext_{kQ}^1(1,2)$. Note that $2$ is a submodule of $\substack{1\\2}$.

Internal merge: so your A and B are different objects with the same subobject D.

Let $Q$ be the quiver $1\to 3 \leftarrow 2$. There are 6 indecomposable module for the corresponding quiver algebra. They look like these combinatorially: $1$, $2$, $3$, $\substack{1 \\ 3}$ and $\substack{2 \\ 3}$, and $\substack{1 \, 2 \\ 3}$. The last diagram really should be V-shaped, one should drew a line (or arrow) in-between 1-3, and 2-3. The last module is an extension of $1$ by $\substack{2\\3}$, or extension of $2$ by $\substack{1 \\ 3}$.

Anyway, the point is your A can be represented by $\substack{1 \\ 3}$, B represented by $\substack{2 \\ 3}$, D represented by $3$, the internal merging of A and B is the V-shaped module. Again, gluing is explained by extension of modules.

I don't understand what the definition of parallel merging is saying; so I will leave it to you to play around with this diagram combinatorics.

Answer 2

Maybe this excerpt from the book "Doing Mathematics" by Martin H.Krieger will help:

“Homological algebra…started in fact as a kind of glorified linear algebra [matrices], by introducing concepts such as the Ext and Tor functors, which in a way measure the manner in which modules over general rings misbehave [that is, just where division may not be possible] when compared to the nice vector spaces of classical linear algebra; and the similarity with homology groups, which tell us how much a complex deviates from being acyclic [or hole free], is now a commonplace.”

Jean Dieudonné, “Recent Developments in Mathematics”, American Mathematical Monthly 71 (1964), p.239-248. The quote corresponds to pages 243-244. Further on, one may read: “The introduction of sheaves by Leray… [gave]a workable mathematical formalism for the intuitive concept of “variation”of structures…doing away with the cumbersome triangulations of former methods…” So the algebraic and analytic notions displace the combinatorial ones.