
4
It seems that equivalence of filters on different bases can be generalized: fil-
ters A ∈ A and B ∈ B are equivalent iff there exists an X ∈ A ∩ B which is greater
than both A and B. This however works only in the case if order of the orders A
and B agree, that is if then are both a suborders of a greater fixed order.
Under which conditions a function spaces of posets is strongly separable?
Generalize both funcoids and reloids as filters on a superset of the lattice Γ (see
“Funcoids are filters” chapter).
When the set of filters closed regarding a funcoid is a (co-)frame?
If a formula F (x0, . . . , xn) holds for every poset Ai then it also holds for product
order
Q
A. (What about infinite formulas like complete lattice joins and meets?)
Moreover F (x0, . . . , xn) = λi ∈ n : F (x0, i, . . . , xn, i) (confused logical forms and
functions). It looks like a promising approach, but how to define it exactly? For
example, F may be a form always true for boolean lattices or for Heyting lattices,
or whatsoever. How one theorem can encompass all kinds of lattices and posets?
We may attempt to restrict to (partial) functions determined by order. (This is not
enough, because we can define an operation restricting \ defined only for posets
of cardinality above or below some cardinal κ. For such restricted \ the above
formula does not work.) See also https://portonmath.wordpress.com/2016/01/
12/a-conjecture-about-product-order-and-logic/. It seems that Todd Trimble
shows a general category-theoretic way to describe this: https://nforum.ncatlab.
org/discussion/6887/operations-on-product-order/.
Get results from http://ncatlab.org/toddtrimble/published/topogeny.
What about distributivity of quasicomplements over meets and joins for the
filtrator of funcoids? Seems like nontrivial conjectures.
Conjecture: Each filtered filtrator is isomorphic to a primary filtrator. (If it
holds, then primary and filtered filtrators are the same!)
Add analog of the last item of the theorem about co-complete funcoids for point-
free funcoids.
Generalize theorems about RLD(A; B) as F (A × B) in order to clean up the
notation (for example in the chapter “Funcoids are filters”).
Define reloids as a filtrator whose core is an ordered semigroup. This way reloids
can be described in several isomorphic ways (just like primary filtrators are both
filtrators of filters, of ideals, etc.) Is it enough to describe all properties of reloids?
Well, it is not a semigroup, it is a precategory. It seems that we also need functions
dom and im into partially ordered sets and “reversion” (dagger).
http://mathoverflow.net/a/191381/4086 says that n-staroids can be identified
with certain ideals!
To relax theorem conditions and definition, we can define protofuncoids as arbi-
trary pairs (α; β) of functions between two posets. For protofuncoids composition
and reverse are defined.
Add examples of funcoids to demonstrate their power: D t T (D is a digraph T
is a topological space), T u
n
(x;y)
y≥x
o
as “one-side topology” and also a circle made
from its π-length segment.
Say explicitly that pseudodifference is a special case of difference.
For pointfree funcoids, if f : A → B exists, then existence of least element of A
is equivalent to existence of least element of B: y 6 hfi⊥
A
⇔ ⊥
A
6 hf
−1
iy ⇔ 0.
Thus hf iy hfiy and so hf iy = ⊥
B
. Can a similar statement be made that A being
join-semilattice implies B being join-semilattice (at least for separable posets)? If