Re: Comments on Norbert's topological extension of relational algebra

From: Norbert_Paul <norbertpauls_spambin_at_yahoo.com>
Date: Tue, 15 Dec 2015 19:53:45 +0100
Message-ID: <n4pnf2$26f$1_at_dont-email.me>

Jan Hidders wrote:
> Op maandag 14 december 2015 20:36:52 UTC+1 schreef Tegiri Nenashi:

>> On Saturday, December 12, 2015 at 4:29:41 AM UTC-8, Norbert_Paul wrote:
>>> Tegiri Nenashi wrote:
>>>> On Friday, December 11, 2015 at 10:33:48 AM UTC-8, Tegiri Nenashi wrote:
>>>>> A [binary] relation induces Galois Connection and closure operator. It doesn't necessarily satisfy union axiom
>>>>> http://mathoverflow.net/questions/35719/when-does-a-galois-connection-induce-a-topology
>>>>> There are several ways to generate a topology from given binary relation
>>>>> http://www.emis.de/journals/BMMSS/pdf/v31n1/v31n1p4.pdf
>>>>
>>>> Perhaps people are bewildered what Galois Connection has to do with it. One particular way by which Galois connections often arise is relation-generated. It involves posets -- powersets of some set X and Y, induced by a relation R between the elements of the original X and Y. Galois's original example
>>>> was of this kind. Wille's Formal Concept Analysis is about relation-generated Galois Connections too.
>>>>
>>>> Now what about closure operator induced by relation-generated Galois Connection? It is still not necessarily Kuratovski closure. Counterexample:
>>>
>>> Topological closure is always Kuratovski closure.
>>> So Galois-connection may be of theoretical (and maybe
>>> even practical) interest but it is not part of my model.
>>
>> However it is interesting to know how much of topological structure can be taken away (generalized) and the result would still be compatible with relational algebra.
>>
>> True or false?
>> Proposition. Let σ : 2^X → 2^X be a map (where 2^X is a powerset of X). Then, the following statements are equivalent.
>> (1) σ is topological closure operator on X,
>> (2) σ satisfies Kuratovski axiom, and there is a Galois connection (K, L) which satisfies σ = LK.
>

> False, I think. The identity would satisfy always (2) but not necessarily (1).
I do not know much about Galois connections, but if identity can be constructed from a Galois conenction in the above manner then it is not a counter-example. Identity is the closure operator of the discrete topological space.

> What *I* am interested in is the connection with this work:

>

> http://alpha.uhasselt.be/~lucp1080/queries_reals.pdf
>
> :-)

Looks interesting. I'll take a closer look. Received on Tue Dec 15 2015 - 19:53:45 CET