"Transitive closure"-like relation [Math]

From: In-Betweener on 7 Jul 2010 17:57

Em 07/07/2010 05:31, William Elliot wrote:
> On Tue, 6 Jul 2010, In-Betweener wrote:
> (...)
>
> Exercise. R is the union of the transitive interiors of R.
>
> ----

Thanks to all of you. Weeks ago, I found that such a transitive interior
was not unique, but after choosing one of them for a particular
relation, I got so impressed with the resemblance of concepts that I
forgot totally that I was not dealing with a unique "interior". Shame on
me. :-)

By the way, I avoided the intersection definition of R+ because the
hypothetical R- couldn't be defined this way, nor using union, of course.

The lattice structure explained by William enriches a lot the range of
insights I can have with my "transitive toys".

Best regards.

From: In-Betweener on 31 Jul 2010 10:17

Em 06/07/2010 19:59, In-Betweener wrote:
> (...)

Hi all.

I'm back after some weeks off-line.

In the meanwhile, I worked a little on the "transitive interior" concept.

My main interest in the concept was on how to keep the transitivity of a
relation after removing an arbitrary pair of it. By the way, I've found
a work akin to the subject at
http://www.cs.uu.nl/research/techreps/repo/CS-1987/1987-25.pdf, but it
concentrates on transitive closure and *reduction*, besides to be
concerned mostly about efficiency.

Here are some results I found myself. Suppose R is a transitive relation
and S is R - {<a,b>}. It is provable that S has at most two "transitive
interiors", defined as below:

T1 = S \ {<a,y> | ySb}
T2 = S \ {<x,b> | aSx}

Additionally, it is simple to define a third, unique, transitive
relation, intersecting the two above:

T = T1 /\ T2

I chose to call T *the* "transitive core" of S. For an arbitrary
relation R, the transitive core, possibly empty, is the intersection of
all transitive interiors of R. If R is already transitive, the
transitive closure, the (unique, in this case) transitive interior and
the transitive core are all the same.

I would appreciate any comments or suggestions about the matter and the
proposed terminology. I've found nothing on the subject so far.

Thanks in advance.