![](/files/happy5.png)
Porton, Victor
A construction of direct product in the category of continuous maps between endo-funcoids ★★★
Author(s): Porton
Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids.
Remind from my book that morphisms of this category are defined by the formula
(here and below by abuse of notation I equate functions with corresponding principal funcoids).
Let are endofuncoids,
We define
(here and
are cartesian projections).
Keywords: categorical product; direct product
Distributivity of a lattice of funcoids is not provable without axiom of choice ★
Author(s): Porton
![$ \mathsf{FCD}(A;B) $](/files/tex/d051d7da40c4a6b1d4234c0f74689d1bc7c994f1.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ B $](/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png)
A similar conjecture:
![$ a\setminus^{\ast} b = a\#b $](/files/tex/c6fc3b6da0655ddeaeffe670703a33edcb4650f6.png)
![$ a $](/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png)
![$ b $](/files/tex/b94226d9717591da8122ae1467eda72a0f35d810.png)
Keywords: axiom of choice; distributive lattice; distributivity; funcoid; reverse math; reverse mathematics; ZF; ZFC
Values of a multifuncoid on atoms ★★
Author(s): Porton
![$ 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 $](/files/tex/20328c795890b2a043f28afc705aecd5679f72d9.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
Keywords:
A conjecture about direct product of funcoids ★★
Author(s): Porton
![$ f_1 $](/files/tex/472cbb02b9334d02f91ee6d85190987d38d55395.png)
![$ f_2 $](/files/tex/0470e2ed22fecd48d8ad613016446ca8d5540085.png)
![$ \operatorname{Src}f_1=\operatorname{Src}f_2=A $](/files/tex/74804e1f464f0ed9acb65fbbe676a40dc27a919b.png)
![$ f_1 \times^{\left( D \right)} f_2 $](/files/tex/0a40b9e9b410bc22dbd5b368fbadae778feb7e62.png)
![$ x $](/files/tex/e7ba5befcaa0d78e43b5176d70ce67425fd0fcdc.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$](/files/tex/2326592ad6b621142c337b6acc2a4b724ca723f4.png)
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
![$ F $](/files/tex/bfff269cc7df9bdb7c57d8b6a2a74020d114f24d.png)
![$ F_i $](/files/tex/468bb283d323c2779688c7954a9de964d909524f.png)
![$ \lambda j \in N \left( i \right) : \mathfrak{F} \left( U_j \right) $](/files/tex/ec2677105f186d63c73702433a4fe09bcc1e92bc.png)
![$ N \left( i \right) $](/files/tex/ccbc2dd2c65699e06cafe73a1ded06b193ab52f0.png)
![$ i $](/files/tex/ca49c241ece07915c97a31774a977841c6f0414c.png)
![$ U_j $](/files/tex/246b680371106f172aa027bc2c18e977aa2c779b.png)
![$ j $](/files/tex/282a1b4a166eb4c8018038a9ca0d509b2c90b79b.png)
![$ F_i = E^{\ast} f_i $](/files/tex/2e6c4a540867fd2ef352ea31f20e7d00c81d8b07.png)
![$ f_i $](/files/tex/3d8d4274a2a88f65b70123aab5f541cbfb114eb2.png)
![$ \lambda j \in N \left( i \right) : \mathfrak{P} \left( U_j \right) $](/files/tex/0154473bda09e018c819cbf7c181386972c20d4a.png)
![$ \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) $](/files/tex/3be7158866e88b56c057bdb5d058cd619d47bdd4.png)
![$ H $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
![$ F $](/files/tex/bfff269cc7df9bdb7c57d8b6a2a74020d114f24d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ Z $](/files/tex/b85ee8eb04b7b36a6f05b7f6218bef305019f707.png)
![$ h $](/files/tex/cc23d2b0c281befd62457dcbedbf9385be2cd2f6.png)
![$ \lambda j \in Z : \mathfrak{P} \left( U_j \right) $](/files/tex/df5585afb153a7713c3b972518a8808361951cf0.png)
![$ H = E^{\ast} h $](/files/tex/58f97b913328b408982a64ecf4dd15fda6ad2df8.png)
![$ \left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right) $](/files/tex/5a6d012c147b71bceeaa2eb8bf011efb3caa8ca5.png)
Keywords: graph-product; multifuncoid
Atomicity of the poset of multifuncoids ★★
Author(s): Porton
![$ (\mathscr{P}\mho)^n $](/files/tex/02023fd10859168be6be125aa8d3912904f57a27.png)
![$ \mho $](/files/tex/8a60ffddf5f015b5658f273c0698af378fdebbde.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
- \item atomic; \item atomistic.
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
Keywords: multifuncoid
Atomicity of the poset of completary multifuncoids ★★
Author(s): Porton
![$ (\mathscr{P}\mho)^n $](/files/tex/02023fd10859168be6be125aa8d3912904f57a27.png)
![$ \mho $](/files/tex/8a60ffddf5f015b5658f273c0698af378fdebbde.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
- \item atomic; \item atomistic.
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
Keywords: multifuncoid
Upgrading a completary multifuncoid ★★
Author(s): Porton
Let be a set,
be the set of filters on
ordered reverse to set-theoretic inclusion,
be the set of principal filters on
, let
be an index set. Consider the filtrator
.
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \mathfrak{P}^n $](/files/tex/81d0b4bc571faee2e8f352e99db8b54be6f9bb4f.png)
![$ E^{\ast} f $](/files/tex/93b48164cf5af924121c451b6bb17268ae140bae.png)
![$ \mathfrak{F}^n $](/files/tex/34e3ea9e5283eb80b19513453991c07af9c98f8a.png)
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
Keywords:
Funcoidal products inside an inward reloid ★★
Author(s): Porton
![$ a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f $](/files/tex/0435b975c85d609a344bf2dcbd5cfacf4cccef40.png)
![$ a \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} b \subseteq f $](/files/tex/540c8fb9c1b2fbbe12b46d35f6cb7aa3511a989b.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ a $](/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png)
![$ b $](/files/tex/b94226d9717591da8122ae1467eda72a0f35d810.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
A stronger conjecture:
![$ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensuremath{\operatorname{in}}} f $](/files/tex/dbf9c288bef784da3d3a5d335b1ca97c01219f4e.png)
![$ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{FCD}}}} \mathcal{B} \subseteq f $](/files/tex/a4b67c268d0f2a939ae1d63a31fd57a3945e4ec5.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \mathcal{A} \in \mathfrak{F} \left( \ensuremath{\operatorname{Src}}f \right) $](/files/tex/52f123c93b14e31b514495ac39d8781cd597c443.png)
![$ \mathcal{B} \in \mathfrak{F} \left( \ensuremath{\operatorname{Dst}}f \right) $](/files/tex/7310f0f823acdb77df75af0f8c924cdd8df0fe53.png)
Keywords: inward reloid
Distributivity of inward reloid over composition of funcoids ★★
Author(s): Porton
Keywords: distributive; distributivity; funcoid; functor; inward reloid; reloid
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