Date: Oct 9, 2013 4:13 AM
Author: William Elliot
Subject: Product, Filters and Quantales

First a proposition about binary relations

If C and D are collections of binary relations
C subset P(XxY), D subset P(Y,Z)
then (using o for composition)
\/C o \/D = \/{ RoS | R in C, S in D }

If F is a filter for XxY and G a filter for YxZ, then the
composition of F and G, F o G is the filter generated by
the filter base
B(F,G) = { AoB | A in F, G in B }.

From the above lemma, we can prove the theorem that if

for all j in J, k in K, Fj filter for XxY, Gk filter for YxZ,
then /\_j Fj o /\_k Gk = /\{ Fj o Gk | j in J, k in K }

Additionally, composition can be proved to be associative.

Since the intersection of filters is the largest filter smaller than
all the filters of the intersection, the conclusion can be restated
inf_j Fj o inf_k Gk = inf{ Fj o Gk | j in J, k in K }

In particular, for F a filter for XxY, G a filter for YxZ
F o inf_k Gk = inf{ F o Gk | k in K }
inf_j Fj o G = inf{ Fj o G | j in J }


For filters of products to be closed under composition,
it's necessary to limit the collection to filters for XxX.

Thusly Ft, the collection of filters for XxX with composition o,
is a semigroup (Ft,o). In fact it's a monoid for the principal
filter generated by D_X = { (x,x) | x in X ) is an identity for
(Ft,o)

In addition, as Ft is ordered by subset, the addition of the
singular filter P(XxX), the filter with the empty set, Ft is a
complete order or complete lattice of filters.


Upon examination, it's seen that (Ft,o,subset) is an order dual
of a quantale, a unital quantale. If, on the other hand, the
order of Ft is reversed the corollary
F o inf_k Gk = inf{ F o Gk | k in K }
inf_j Fj o G = inf{ Fj o G | j in J }
becomes
F o sup_k Gk = sup{ F o Gk | k in K }
sup_j Fj o G = sup{ Fj o G | j in J }

Thusly (Ft,o,reversed subset) is a unital quantale.

Any comments, additions, corrections, counter examples
questions or requests for proofs?

Now a filter F, for XxX is considered an uniformity, making
X a uniform space. In topological or uniform spaces concepts
or terminology, how would one describe the composition of two
uniformities for the same set? Is composition of uniformities
a useful notion?