![](/files/happy5.png)
Porton, Victor
Funcoid corresponding to reloid through lattice Gamma ★★
Author(s): Porton
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \mathcal{X} \in \mathfrak{F} (\operatorname{Src} f) $](/files/tex/1f01dd9f1243b55507248bea7af215e6469c00a8.png)
![$ \mathcal{Y} \in \mathfrak{F} (\operatorname{Dst} f) $](/files/tex/d7fa63ea4e0a3ad7a7f39a92a95ae4fe1906197d.png)
- \item
![$ \mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y} \Leftrightarrow \forall F \in \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y} $](/files/tex/2f0c7dbaa1a5747d9bca753501374e8cd2500318.png)
![$ \langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigcap_{F \in \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f} \langle F \rangle \mathcal{X} $](/files/tex/bcc925b41f94370c0dfe32107235c7a3435dcbf9.png)
It's proved by me in this online article.
Keywords: funcoid corresponding to reloid
Restricting a reloid to lattice Gamma before converting it into a funcoid ★★
Author(s): Porton
![$ (\mathsf{FCD}) f = \bigcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \operatorname{GR} f) $](/files/tex/d9afe4920809a29a644f5bc594e40f3313a8d527.png)
![$ f \in \mathsf{RLD} (A ; B) $](/files/tex/90326a901389c8760f7fa928fff117636a958338.png)
Keywords: funcoid corresponding to reloid; funcoids; reloids
Inner reloid through the lattice Gamma ★★
Author(s): Porton
![$ (\mathsf{RLD})_{\operatorname{in}} f = \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $](/files/tex/9c5b448dbc0964ca844d30e92247626e8d5420b5.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
Counter-example: for the funcoid
is proved in this online article.
Keywords: filters; funcoids; inner reloid; reloids
Coatoms of the lattice of funcoids ★
Author(s): Porton
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ B $](/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png)
![$ \mathsf{FCD}(A;B) $](/files/tex/d051d7da40c4a6b1d4234c0f74689d1bc7c994f1.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ B $](/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png)
![$ \mathsf{FCD}(A;B) $](/files/tex/d051d7da40c4a6b1d4234c0f74689d1bc7c994f1.png)
Direct proof of a theorem about compact funcoids ★★
Author(s): Porton
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ T_1 $](/files/tex/d9a989037d7243d9036b9c8165c72e0331991a8c.png)
![$ T_2 $](/files/tex/55e29109946a61a46e41f972a62209d3dbd4e96c.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$ ( \mathsf{\tmop{FCD}}) g = f $](/files/tex/f68984666c5a1553c57c071dd482b9e3f1869eb4.png)
![$ g = \langle f \times f \rangle^{\ast} \Delta $](/files/tex/630751f9d9f5276b67a64fc57d858c975ce7f9e4.png)
The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.
The direct proof may be constructed by correcting all errors an omissions in this draft article.
Direct proof could be better because with it we would get a little more general statement like this:
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ T_1 $](/files/tex/d9a989037d7243d9036b9c8165c72e0331991a8c.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
- \item
![$ ( \mathsf{\tmop{FCD}}) g = f $](/files/tex/f68984666c5a1553c57c071dd482b9e3f1869eb4.png)
![$ g \circ g^{- 1} \sqsubseteq g $](/files/tex/128f5af4e5b7cfd743bb0eb4fe454e040407e28f.png)
Then .
Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity
Generalized path-connectedness in proximity spaces ★★
Author(s): Porton
Let be a proximity.
A set is connected regarding
iff
.
![$ \mu $](/files/tex/12e00f6f7e80e7b1fd1b89a31a7e0abe4c1b1302.png)
![$ U $](/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png)
- \item
![$ U $](/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png)
![$ \mu $](/files/tex/12e00f6f7e80e7b1fd1b89a31a7e0abe4c1b1302.png)
![$ a, b \in U $](/files/tex/b9afee507ee1fdd9052fada0ecf84737e5f57da8.png)
![$ P \subseteq U $](/files/tex/6bd29b775b103849d9f752b5b7bcb156d949341f.png)
![$ \min P = a $](/files/tex/823bbdacb0b048c6f8f9e75c63e39794dd918e62.png)
![$ \max P = b $](/files/tex/3def450ce9e9804c9d4a8be3ece00739a3893a85.png)
![$ \{ X, Y \} $](/files/tex/69a8a2715ddfa2480ba4a8321219c905c5402338.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ X $](/files/tex/302cdeba125e821f3406302c9789229d48f42ea7.png)
![$ Y $](/files/tex/6e8160788c99301d68bd6cf12fcc0ed07fd138d7.png)
![$ \forall x \in X, y \in Y : x < y $](/files/tex/4b87461ca127a265ebe9b2b7dcf0ccbed211c81e.png)
![$ X \mathrel{[ \mu]^{\ast}} Y $](/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png)
Keywords: connected; connectedness; proximity space
Every monovalued reloid is metamonovalued ★★
Author(s): Porton
Keywords: monovalued
Every metamonovalued reloid is monovalued ★★
Author(s): Porton
Keywords:
Every metamonovalued funcoid is monovalued ★★
Author(s): Porton
The reverse is almost trivial: Every monovalued funcoid is metamonovalued.
Keywords: monovalued
Decomposition of completions of reloids ★★
Author(s): Porton
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
- \item
![$ \operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f $](/files/tex/0844704618d467ae0507a89bdb4b215b28d57759.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \operatorname{CoCompl} ( f \circ g) = f \circ \operatorname{CoCompl} g $](/files/tex/91a46d5da19358329c8cdb7351eae9cdb03c2764.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \operatorname{CoCompl} ( ( \operatorname{Compl} g) \circ f) = \operatorname{Compl} ( g \circ ( \operatorname{CoCompl} f)) = ( \operatorname{Compl} g) \circ ( \operatorname{CoCompl} f) $](/files/tex/92e3ac66e1ae6505f78e9f443665d1bcb234fe13.png)
![$ \operatorname{Compl} ( g \circ ( \operatorname{Compl} f)) = \operatorname{Compl} ( g \circ f) $](/files/tex/528dc0c7455fad4558f8b970c989e96990663021.png)
![$ \operatorname{CoCompl} ( ( \operatorname{CoCompl} g) \circ f) = \operatorname{CoCompl} ( g \circ f) $](/files/tex/61a4422f23782d6433a39d5ed6a48dd554f3d16f.png)
Keywords: co-completion; completion; reloid
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