Generalized Fréchet filters
Just a few minutes ago I conceived a definition of generalized Fréchet filters with definition for every poset on which filters are considered (however, I have not yet calculated the class of posets for which generalized Fréchet filter is defined; it should be easy but I am busy with other business). Generalized Fréchet filter on […]
Another star-category of funcoids
I’ve introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible, because I have failed to prove it. This should be included in the next version of my book.
The set of funcoids is a co-frame (without axiom of choice)
A mathematician named Todd Trimble has helped me to prove that the set of funcoids between two given sets (and more generally certain pointfree funcoids) is always a co-frame. (I knew this for funcoids but my proof required axiom of choice, while Todd’s does not require axiom of choice.) He initially published his proof here […]
A conjecture about multifuncoids and ultrafilters is proved
I’ve proved the following conjecture: Theorem Let $latex f$ be a staroid such that $latex (\mathrm{form}\, f)_i$ is an atomic lattice for each $latex i \in \mathrm{arity}\, f$. We have $latex \displaystyle L \in \mathrm{GR}\, f \Leftrightarrow \mathrm{GR}\, f \cap \prod_{i \in \mathrm{dom}\, \mathfrak{A}} \mathrm{atoms}\, L_i \neq \emptyset $ for every $latex L \in \prod_{i […]
Category theoretical generalization of reloids and funcoids
While walking home from McDonalds I conceived the following idea how we can generalize reloids and funcoids. Let $latex C$ be a category with finite products, the set of objects of which is a complete lattice (for the case of funcoids as described below it is enough to be just join-semilattice). One can argue which […]
I am going to rewrite my research monograph
When my book was already sent to a publisher, I decided to rewrite it. Here is my rewriting plan (what I am going to change in the book).
Error correction: definition of multifuncoids was wrong
I have corrected some errors in my book related to my definition of multifuncoids. The definition of multifuncoid was wrong and this triggered several errors along my book.
I added a new chapter to my book
After checking for errors I added (as a new chapter) materials of the article Identity staroids into my research monograph. I plan to “dissolve” this chapter, that is distribute its materials among other chapters and liquidate this chapter itself.
Draft article about identity staroids
I’ve mostly finished writing the article Identity Staroids which considers $latex n$-ary identity staroids (with possibly infinite $latex n$), which generalize $latex n$-ary identity relations and some related topics. (In my theory there are two kinds of identity staroids: big and small identity staroids.) Writing of the article is mostly finished, I am going just […]
Products in dagger categories – updated
I have rewritten my draft article Products in dagger categories with complete ordered Mor-sets. Now I denote the product of an indexed family $latex X$ of objects as $latex \prod^{(Q)} X$ (instead of old confusing $latex Z’$ and $latex Z”$ notation) and the infimum product and supremum coproduct correspondingly as $latex \prod^{(L)} X$ and $latex […]