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.
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 […]
The set of ideals on an infinite join-semilattice is a boolean algebra
The below is wrong, because pointfree funcoids between boolean algebras are not the same as 2-staroids between boolean algebras. It was an error. I have just discovered that the set of ideals on an infinite join-semilattice is a boolean algebra (moreover it is a complete atomistic boolean algebra). For me, it is a very counter-intuitive […]
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 […]
Pointfree reloids discovered
After I defined pointfree funcoids which generalize funcoids (see my draft book) I sought for pointfree reloids (a suitable generalization of reloids, see my book) long time. Today I have finally discovered pointfree reloids. The idea is as follows: Funcoids between sets $latex A$ and $latex B$ denoted $latex \mathsf{FCD}(A;B)$ are essentially the same as […]
Binary relations are essentially the same as pointfree funcoids between powersets
After this Math.StackExchange question I have proved that binary relations are essentially the same as pointfree funcoids between powersets. Full proof is available in my draft book. The most interesting aspect of this is that is that we can construct filtrator with core being pointfree funcoids from $latex \mathfrak{A}$ to $latex \mathfrak{B}$ for every poset […]
Ideals, free stars, and mixers in my book
I wrote a section on ideals, free stars, and mixers in my book. Now free stars (among with ideals and mixers) are studied as first class objects, being shown isomorphic to filters on posets. In (not so far) future it should allow to extend our research of pointfree funcoids and staroids/multifuncoids, using now comprehensive theory […]