Three (seemingly not so difficult) new conjectures
I’ve noticed the following three conjectures (I expect not very difficult) for finite binary relations $latex X$ and $latex Y$ between some sets and am going to solve them: $latex X\sqcap^{\mathsf{FCD}} Y = X\sqcap Y$; $latex (\top \setminus X)\sqcap^{\mathsf{FCD}} (\top \setminus Y) = (\top \setminus X)\sqcap (\top \setminus Y)$; $latex (\top \setminus X)\sqcap^{\mathsf{FCD}} Y = […]
A new research project (a conjecture about funcoids)
I start the “research-in-the middle” project (an outlaw offspring of Polymath Project) introducing to your attention the following conjecture: Conjecture The following are equivalent (for every lattice $latex \mathsf{FCD}$ of funcoids between some sets and a set $latex S$ of principal funcoids (=binary relations)): $latex \forall X, Y \in S : \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y) […]
Conjecture about funcoids
Conjecture Let $latex S$ be a set of binary relations. If for every $latex X, Y \in S$ we have $latex \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y) \subseteq S$ then there exists a funcoid $latex f$ such that $latex S = \mathrm{up}\, f$.
A new theorem proved
Definition A set $latex S$ of binary relations is a base of a funcoid $latex f$ when all elements of $latex S$ are above $latex f$ and $latex \forall X \in \mathrm{up}\, f \exists T \in S : T \sqsubseteq X$. It was easy to show: Proposition A set $latex S$ of binary relations is […]
I made an error in a proof
I’ve proved the following (for every funcoids $latex f$ and $latex g$): Statement $latex \mathrm{up}\, (f \sqcap^{\mathsf{FCD}} g) \subseteq \bigcup \{ \mathrm{up}\, (F \sqcap^{\mathsf{FCD}} G) \mid F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \}$ or equivalently: If $latex Z\in\mathrm{up}\, (f \sqcap^{\mathsf{FCD}} g)$ then there exists $latex F \in \mathrm{up}\, f$, $latex G \in \mathrm{up}\, […]
I proved a conjecture
After prayer in tongues and going down anointment of Holy Spirit I proved this conjecture about funcoids. The proof is currently located in this PDF file. Well, the proof is for special cases of distributive lattices, but more general case seems not necessary (at least now). It seems easy to generalize it for more general […]
New conjecture about funcoids
New conjecture: Conjecture $latex \mathrm{up} (f \sqcap^{\mathsf{FCD}} g) \subseteq \{ F \sqcap G \mid F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \}$ for all funcoids $latex f$, $latex g$ (with corresponding sources and destinations). Looks trivial? But how to (dis)prove it?
Attempt to generalize filter bases for more general filtrators
In this draft I present some definitions and conjectures on how to generalize filter bases for more general filtrators (such as the filtrator of funcoids). This is a work-in-progress. This seems an interesting research by itself, but I started to develop it as a way to prove this conjecture.
An informal open problem in mathematics
Characterize the set $latex \{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_{\mathrm{out}} f\}$. (This seems a difficult problem.)
A new diagram about funcoids and reloids
Define for posets with order $latex \sqsubseteq$: $latex \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$; $latex \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \}$. Note that the above is a generalization of monotone Galois […]