A diagram about funcoids and reloids ★★

Author(s): Porton

Define for posets with order :

1. ;
2. .

Note that the above is a generalization of monotone Galois connections (with and replaced with suprema and infima).

Then we have the following diagram:

What is at the node "other" in the diagram is unknown.

Conjecture   "Other" is .
Question   What repeated applying of and to "other" leads to? Particularly, does repeated applying and/or to the node "other" lead to finite or infinite sets?

Keywords: Galois connections

Outward reloid of composition vs composition of outward reloids ★★

Author(s): Porton

Conjecture   For every composable funcoids and

Keywords: outward reloid

A funcoid related to directed topological spaces ★★

Author(s): Porton

Conjecture   Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.
Proposition   It is easy to prove that is the infinitely small right neighborhood filter of point .

If proved true, the conjecture then can be generalized to a wider class of posets.

Keywords:

Infinite distributivity of meet over join for a principal funcoid ★★

Author(s): Porton

Conjecture   for principal funcoid and a set of funcoids of appropriate sources and destinations.

Keywords: distributivity; principal funcoid

Entourages of a composition of funcoids ★★

Author(s): Porton

Conjecture   for every composable funcoids and .

Keywords: composition of funcoids; funcoids