SUO: Re: Relations And Their Divisitudes
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RATD. Note 8
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Tom,
Shifting to another thread where I have
all of my asscoiative facilities intact.
I suggested that we begin by checking whether we have
the same concepts of 1-adic and 2-adic projections as
applied to 3-adic relations.
As I read your table I see a possible divergence on this row:
o-----------------------------o---------------------------------------o
| 1-adic projection | result of a relational PROJECT |
| | operation on a table, that |
| | leaves just one column |
o-----------------------------o---------------------------------------o
We are talking about unkeyed tables, right?
Thus, when we extract a column from a table,
only the set of entries counts, not the set
with multiplicites, what some call a "bag",
and certainly not an ordered list of items,
which would amount to a column plus the key.
In other words, I am defining the 1-adic and 2-adic
projections on 3-adic relations in the following way:
The 1-adic "projections" Proj_X, Proj_Y, Proj_Z,
alternatively written as p_1, p_2, p_3,
as applying to a 3-adic relation L c X x Y x Z,
along with the equivalent forms of application
L_X = p_1 (L), L_Y = p_2 (L), L_Z = p_3(L),
respectively, are defined as follows:
Proj_X (L) = L_X = {x in X : <x, y, z> in L for some y in Y, z in Z},
Proj_Y (L) = L_Y = {y in Y : <x, y, z> in L for some x in X, z in Z},
Proj_Z (L) = L_Z = {z in Z : <x, y, z> in L for some x in X, y in Y}.
The 2-adic "projections" Proj_XY, Proj_XZ, Proj_YZ,
alternatively written as p_12, p_13, p_23,
as applying to a 3-adic relation L c X x Y x Z,
along with the equivalent forms of application
L_XY = p_12(L), L_XZ = p_13(L), L_YZ = p_23(L),
respectively, are defined as follows:
Proj_XY (L) = L_XY = {<x, y> in XxY : <x, y, z> in L for some z in Z},
Proj_XZ (L) = L_XZ = {<x, z> in XxZ : <x, y, z> in L for some y in Y},
Proj_YZ (L) = L_YZ = {<y, z> in YxZ : <x, y, z> in L for some x in X}.
Well, it's late, and the risk of typos and worse grows with
this sort of stuff, so I will leave it there for the moment.
Jon Awbrey
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