Funcoids and reloids between arbitrary sets
I decided to rewrite my theory of funcoids and reloids with funcoids and reloids defined between arbitrary sets instead of current theory which describes funcoids and reloids on a fixed set. That way I will make funcoids and reloids into categories with objects being (small) sets and morphisms being funcoids and reloids. That will make […]
“Orderings of filters in terms of reloids” – a formal approach
I updated the draft “Orderings of filters in terms of reloids. Extensions of Rudin-Keisler ordering” at this Web page. In the updated version reloids between different sets (I now call them trans-reloids.) are considered in a formal manner, unlike a somehow informal approach in the previous version of this draft. Instead of proving all properties […]
“Orderings of filters in terms of reloids” – draft updated
I updated the draft “Orderings of filters in terms of reloids. Extensions of Rudin-Keisler ordering” at this Web page. The updated version contains a new theorem (or rather a counter-example) which I proved with the help of Andreas Blass. This version is yet too rough draft and I hope to finish rewriting it in a […]
A new theorem about funcoids and generalizated filter bases
I proved the following theorem: Theorem If $latex S$ is a generalized filter base then $latex \left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S$ for every funcoid $latex f$. The proof (presented in updated version of this online article) is short but not quite trivial. It was originally formulated as […]
A surprisingly hard problem
I am now trying to prove or disprove this innocently looking but somehow surprisingly hard conjecture: Conjecture If $latex S$ is a generalized filter base then $latex \left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S$ for every funcoid $latex f$.
Orderings of filters in terms of reloids – in terms of category theory
My draft article “Orderings of filters in terms of reloids. Extensions of Rudin-Keisler ordering” is updated. Read here. Now it uses the language of basic category theory. It is yet not carefully checked for errors.
Orderings of filters in terms of reloids – online draft
I put a preliminary draft (not yet checked for errors) of the article “Orderings of filters in terms of reloids” at Algebraic General Topology site. The abstract Orderings of filters which extend Rudin-Keisler preorder of ultrafilters are defined in terms of reloids that (roughly speaking) is filters on sets of binary relations between some sets. […]
Two elementary theorems
I proved the following two elementary but useful theorems: Theorem For every funcoids $latex f$, $latex g$: If $latex \mathrm{im}\, f \supseteq \mathrm{im}\, g$ then $latex \mathrm{im}\, (g\circ f) = \mathrm{im}\, g$. If $latex \mathrm{im}\, f \subseteq \mathrm{im}\, g$ then $latex \mathrm{dom}\, (g\circ f) = \mathrm{dom}\, g$. Theorem For every reloids $latex f$, $latex g$: […]
New corollary
To the theorem “Every monovalued reloid is a restricted function.” I added a new corollary “Every monovalued injective reloid is a restricted injection.” See the online article “Funcoids and Reloids” and this Web page.
Compl CoCompl f = CoCompl Compl f = Cor f
I proved true the following conjecture: Theorem $latex \mathrm{Compl} \, \mathrm{CoCompl}\, f = \mathrm{CoCompl}\, \mathrm{Compl}\, f = \mathrm{Cor}\, f$ for every reloid $latex f$. See here for definitions and proofs.