2006/9/23, Roger Hui <[EMAIL PROTECTED]>:
> the rank of the verb determins the cells.
This is not correct. For example, it make sense
to talk about the 1-cells of a matrix (the rows
of a matrix) without reference to any verb.
From your explanation and the dictionary, I suppose that the following
sentence alone:
u/ applies u between each cell of x and the entire y.
does not imply anything about the rank of applying u. This was my
initial confusion when reading it. From the context it read like using
"cell" implied the (left) rank of u.
----- Original Message -----
From: Dan Bron <[EMAIL PROTECTED]>
Date: Friday, September 22, 2006 10:10 am
Subject: Re: Cell (was Re: [Jgeneral] I expect a table of ordered pairs for
this)
> Short answer:
>
> The rank of the noun determines the items,
> the rank of the verb determins the cells.
>
> Every noun can be thought of as a "list of something". For
> example, a vector (a 1 dimensional noun) is a list of scalars, a
> table (2D) is a list of vectors, a cube (3D) is a list of tables,
> and so on. The items of a noun are whatever it's a list of. The
> items of a vector are scalars, the items of a table are vectors,
> the items of a cube are tables, and so on.
>
> Any verb can be limited to a certain rank of noun. That is, it
> can be impossible for a verb to see a noun of higher than a
> specific rank. For example, + can only see scalars, because
> list+list makes no sense: only scalars can be added. So, no
> matter how hard you try, + will only ever see a scalar at a
> time. The highest rank of noun a certain verb can see is a "cell"
> for that verb.
>
> Of course, you can feed a verb a higher ranked argument, in which
> case there will be more than one cell, but the verb will still
> only see one cell at a time. That's why list+list does work:
> it's operating piecewise, adding a scalar at a time.
>
> A verb can have infinite (unbound) rank, in which case ANY
> argument will only have one cell. Think of the monad < (box),
> for example. No matter what argument you give it, it treats that
> argument as an indivisible whole, and returns it in a box.
>
> A verb can also define its cells in terms of the rank of the noun,
> so, for example, "_1 means "operate on rank one less than the
> argument", which means "treat the argument like a list of cells",
> which means "operate on the items of the argument". So, in the
> special case of a verb "_1 , the items and the cells are the same.
>
> Does that clarify?
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