My book was checked for errors
I have checked for errors the entire text of my research monograph Algebraic General Topology. Volume 1 in which I generalize basic concepts of general topology using so called “funcoids” instead of topological spaces. Enjoy reading this prominent math research.
Erroneous theorem turned into a conjecture
Earlier I claimed that I proved the following theorem: $latex (\mathcal{A}\ltimes\mathcal{B})\sqcap(\mathcal{A}\rtimes\mathcal{B})=\mathcal{A}\times_{F}^{\mathsf{RLD}}\mathcal{B}$ for every filters $latex \mathcal{A}$, $latex \mathcal{B}$ on sets. (Here $latex \ltimes$ and $latex \rtimes$ is what I call oblique products.) Now I have found an error in my proof, so now it is presented as a conjecture in my book.
New theorem and conjectures
I have a little generalized the following old theorem: $latex (a\sqcap^{\mathfrak{A}}b)^{\ast}=(a\sqcap^{\mathfrak{A}}b)^{+}=a^{\ast}\sqcup^{\mathfrak{A}}b^{\ast}=a^{+}\sqcup^{\mathfrak{A}}b^{+}$. I have also found a new (easy to prove) theorem: $latex (a\sqcup^{\mathfrak{A}}b)^{\ast}=(a\sqcup^{\mathfrak{A}}b)^{+}=a^{\ast}\sqcap^{\mathfrak{A}}b^{\ast}=a^{+}\sqcap^{\mathfrak{A}}b^{+}$. The above formulas hold for filters on a set (and some generalizations). Do these formulas hold also for funcoids? (an interesting conjecture) See my free e-book.
Pointfree binary relations (a short article)
In a short (4 pages) article I define pointfree binary relations, a generalization of binary relations which does not use “points” (elements). In a certain special case (of endo-relations) pointfree binary relations are essentially the same as binary relations. It seems promising to research filters on sets of pointfree relations, generalizing the notion of reloids […]
A more abstract way to define reloids
We need a more abstract way to define reloids: For example filters on a set $latex A\times B$ are isomorphic to triples $latex (A;B;f)$ where $latex f$ is a filter on $latex A\times B$, as well as filters of boolean reloids (that is pairs $latex (\alpha;\beta)$ of functions $latex \alpha\in (\mathscr{P}B)^{\mathscr{P}A}$, $latex \beta\in (\mathscr{P}B)^{\mathscr{P}A}$ such […]
“Open maps between funcoids” rewritten
There were several errors in the section “Open maps” of my online book. I have rewritten this section and also moved the section below in the book text. However, the new proof of the theorem stating that composition of open maps between funcoids is an open map now uses a proof referring to a particular […]
Pointfree funcoids between join-semilattices conjecture
Today I’ve come up with the following easy to prove theorem (exercise!) for readers of my book: Theorem If there exists at least one pointfree funcoid from a poset $latex \mathfrak{A}$ to a poset $latex \mathfrak{B}$ then either both posets have least element or none of them. This provokes me to the following conjecture also: […]
Regular funcoids, a generalization of regular topospaces (rewritten because of an error)
Both my definition and description of properties of regular funcoids were erroneous. (The definition was not compatible with the customary definition of regular topological spaces due an error in the definition, and its properties included mathematical errors.) I have rewritten the erroneous section of my book. Now it is shown that being regular for a […]
My book is now available free of charge
Today I’ve took the bold decision to put my math research book online free (under Creative Commons license), with LaTeX source available for editing by anyone at a Git hosting. Because of conflict of licensing, it seems not that my book will be never published officially. However publishing in Git has some advantages: If I […]
Another definition of pointfree reloids
In previous post I stated that pointfree reloids can be defined as filters on pointfree funcoids. Now I suggest also an alternative definition of pointfree reloids: Pointfree reloids can be defined as filters on products $latex \mathrm{atoms}\,\mathfrak{A} \times \mathrm{atoms}\,\mathfrak{B}$ of atoms of posets $latex \mathfrak{A}$ and $latex \mathfrak{B}$. In the case if $latex \mathfrak{A}$ and […]