![](/files/happy5.png)
A diagram about funcoids and reloids ★★
Author(s): Porton
Define for posets with order :
;
.
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
.
![$ \lambda f\in\mathsf{FCD}: \top $](/files/tex/4a511edece8921fab6426695d3451efc024273a5.png)
Question What repeated applying of
and
to "other" leads to? Particularly, does repeated applying
and/or
to the node "other" lead to finite or infinite sets?
![$ \Phi_{\ast} $](/files/tex/c26f0d43856d263f335939666a99f483ffd09da8.png)
![$ \Phi^{\ast} $](/files/tex/cf6796f6de9023eb4c0ae3e69b8900a93e53fc6b.png)
![$ \Phi_{\ast} $](/files/tex/c26f0d43856d263f335939666a99f483ffd09da8.png)
![$ \Phi^{\ast} $](/files/tex/cf6796f6de9023eb4c0ae3e69b8900a93e53fc6b.png)
Keywords: Galois connections
Outward reloid of composition vs composition of outward reloids ★★
Author(s): Porton
Conjecture For every composable funcoids
and
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$$(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq(\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}}f.$$](/files/tex/84f69b164792549ff2f890d3c9dfe23addb9c1ac.png)
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.
![$ R $](/files/tex/201b5ff8bf9045c34a583adc2741b00adf1fd14c.png)
![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
![$ \geq $](/files/tex/45f96d07de2ad307ec6b9d5fbad7c02d93d9eaf2.png)
![$ R\sqcap^{\mathsf{FCD}}\mathord{\geq} $](/files/tex/5521c999ae08fc16a7a797a3fd66316435ad7aff.png)
Proposition It is easy to prove that
is the infinitely small right neighborhood filter of point
.
![$ \langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\} $](/files/tex/4a22ece277f13be752937ec312efed1484d5d2b8.png)
![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
If proved true, the conjecture then can be generalized to a wider class of posets.
Keywords:
Infinite distributivity of meet over join for a principal funcoid ★★
Author(s): Porton
Conjecture
for principal funcoid
and a set
of funcoids of appropriate sources and destinations.
![$ f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S $](/files/tex/0a7e06f88b6cd4667f7fa4b6f670b57cfa795155.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
Keywords: distributivity; principal funcoid
Entourages of a composition of funcoids ★★
Author(s): Porton
Conjecture
for every composable funcoids
and
.
![$ \forall H \in \operatorname{up} (g \circ f) \exists F \in \operatorname{up} f, G \in \operatorname{up} g : H \sqsupseteq G \circ F $](/files/tex/145d32f0a6448ea3d98f9b091370160956ba4532.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
Keywords: composition of funcoids; funcoids