Porton, Victor


Distributivity of a lattice of funcoids is not provable without axiom of choice

Author(s): Porton

Conjecture   Distributivity of the lattice $ \mathsf{FCD}(A;B) $ of funcoids (for arbitrary sets $ A $ and $ B $) is not provable in ZF (without axiom of choice).

A similar conjecture:

Conjecture   $ a\setminus^{\ast} b = a\#b $ for arbitrary filters $ a $ and $ b $ on a powerset cannot be proved in ZF (without axiom of choice).

Keywords: axiom of choice; distributive lattice; distributivity; funcoid; reverse math; reverse mathematics; ZF; ZFC

Values of a multifuncoid on atoms ★★

Author(s): Porton

Conjecture   $ L \in \mathrel{\left[ f \right]} \Rightarrow \mathrel{\left[ f \right]} \cap \prod_{i \in \operatorname{dom} \mathfrak{A}} \operatorname{atoms} L_i \neq \emptyset $ for every multifuncoid $ f $ of the form whose elements are atomic posets.

Keywords:

A conjecture about direct product of funcoids ★★

Author(s): Porton

Conjecture   Let $ f_1 $ and $ f_2 $ are monovalued, entirely defined funcoids with $ \operatorname{Src}f_1=\operatorname{Src}f_2=A $. Then there exists a pointfree funcoid $ f_1 \times^{\left( D \right)} f_2 $ such that (for every filter $ x $ on $ A $) $$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$ (The join operation is taken on the lattice of filters with reversed order.)

A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.

Keywords: category theory; general topology

Graph product of multifuncoids ★★

Author(s): Porton

Conjecture   Let $ F $ is a family of multifuncoids such that each $ F_i $ is of the form $ \lambda j \in N \left( i \right) : \mathfrak{F} \left( U_j \right) $ where $ N \left( i \right) $ is an index set for every $ i $ and $ U_j $ is a set for every $ j $. Let every $ F_i = E^{\ast} f_i $ for some multifuncoid $ f_i $ of the form $ \lambda j \in N \left( i \right) : \mathfrak{P} \left( U_j \right) $ regarding the filtrator $ \left( \prod_{j \in N \left( i \right)} \mathfrak{F} \left( U_j \right) ; \prod_{j \in N \left( i \right)} \mathfrak{P} \left( U_j \right) \right) $. Let $ H $ is a graph-composition of $ F $ (regarding some partition $ G $ and external set $ Z $). Then there exist a multifuncoid $ h $ of the form $ \lambda j \in Z : \mathfrak{P} \left( U_j \right) $ such that $ H = E^{\ast} h $ regarding the filtrator $ \left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right) $.

Keywords: graph-product; multifuncoid

Atomicity of the poset of multifuncoids ★★

Author(s): Porton

Conjecture   The poset of multifuncoids of the form $ (\mathscr{P}\mho)^n $ is for every sets $ \mho $ and $ n $:
    \item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords: multifuncoid

Atomicity of the poset of completary multifuncoids ★★

Author(s): Porton

Conjecture   The poset of completary multifuncoids of the form $ (\mathscr{P}\mho)^n $ is for every sets $ \mho $ and $ n $:
    \item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords: multifuncoid

Upgrading a completary multifuncoid ★★

Author(s): Porton

Let $ \mho $ be a set, $ \mathfrak{F} $ be the set of filters on $ \mho $ ordered reverse to set-theoretic inclusion, $ \mathfrak{P} $ be the set of principal filters on $ \mho $, let $ n $ be an index set. Consider the filtrator $ \left( \mathfrak{F}^n ; \mathfrak{P}^n \right) $.

Conjecture   If $ f $ is a completary multifuncoid of the form $ \mathfrak{P}^n $, then $ E^{\ast} f $ is a completary multifuncoid of the form $ \mathfrak{F}^n $.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords:

Funcoidal products inside an inward reloid ★★

Author(s): Porton

Conjecture   (solved) If $ a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f $ then $ a \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} b \subseteq f $ for every funcoid $ f $ and atomic f.o. $ a $ and $ b $ on the source and destination of $ f $ correspondingly.

A stronger conjecture:

Conjecture   If $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f $ then $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} \mathcal{B} \subseteq f $ for every funcoid $ f $ and $ \mathcal{A} \in \mathfrak{F} \left( \ensuremath{\operatorname{Src}}f \right) $, $ \mathcal{B} \in \mathfrak{F} \left( \ensuremath{\operatorname{Dst}}f \right) $.

Keywords: inward reloid

Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

Conjecture   $ ( \mathsf{\tmop{RLD}})_{\tmop{in}} (g \circ f) = ( \mathsf{\tmop{RLD}})_{\tmop{in}} g \circ ( \mathsf{\tmop{RLD}})_{\tmop{in}} f $ for any composable funcoids $ f $ and $ g $.

Keywords: distributive; distributivity; funcoid; functor; inward reloid; reloid

Upgrading a multifuncoid ★★

Author(s): Porton

Let $ \mho $ be a set, $ \mathfrak{F} $ be the set of filters on $ \mho $ ordered reverse to set-theoretic inclusion, $ \mathfrak{P} $ be the set of principal filters on $ \mho $, let $ n $ be an index set. Consider the filtrator $ \left( \mathfrak{F}^n ; \mathfrak{P}^n \right) $.

Conjecture   If $ f $ is a multifuncoid of the form $ \mathfrak{P}^n $, then $ E^{\ast} f $ is a multifuncoid of the form $ \mathfrak{F}^n $.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords:

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