New version of my math research book
I’ve released a new version of my free math ebook. The main feature of this new release is chapter “Alternative representations of binary relations” where I essentially claim that the following are the same: binary relations pointfree funcoids between powersets Galois connections between powersets antitone Galois connections between powersets This theorem is presented with a […]
A new math abstraction, categories of sides
I introduce a new math abstraction, categories of sides, in order to generalize two theorems into one. Category of sides $latex \Upsilon$ is an ordered category whose objects are (small) bounded lattices and whose morphisms are maps between lattices such that every Hom-set is a bounded lattice and (for all relevant variables): $latex (a \sqcup […]
A new negative result in pointfree topology
I have proved the following negative result: Theorem $latex \mathsf{pFCD} (\mathfrak{A};\mathfrak{A})$ is not boolean if $latex \mathfrak{A}$ is a non-atomic boolean lattice. The theorem is presented in this file. $latex \mathsf{pFCD}(\mathfrak{A};\mathfrak{B})$ denotes the set of pointfree funcoids from a poset $latex \mathfrak{A}$ to a poset $latex \mathfrak{B}$ (see my free ebook). The theorem and its […]
Galois connections are related with pointfree funcoids!
I call pointfree funcoids (see my free e-book) between boolean lattices as boolean funcoids. I have proved that: Theorem Let $latex \mathfrak{A}$ and $latex \mathfrak{B}$ be complete boolean lattices. Then $latex \alpha$ is the first component of a boolean funcoid iff it is a lower adjoint (in the sense of Galois connections between posets). Does […]
I’ve partially proved a conjecture
The following is a conjecture: Conjecture The set of pointfree funcoids between two boolean lattices is itself a boolean lattice. Today I have proved its special case: Theorem The set of pointfree funcoids between a complete boolean lattice and an atomistic boolean lattice is itself a boolean lattice. It is a very weird theorem because […]
My math research monograph updated
I have uploaded a new version of my research monograph in general topology. It weakens conditions of some theorems in “Pointfree funcoids” section (thus making theorems more general), restructures the text and contain other small changes. The book download is freely available.
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.
A conjecture about product order and logic
The considerations below were with an error, see the comment. Product order $latex {\prod \mathfrak{A}}&fg=000000$ of posets $latex {\mathfrak{A}_i}&fg=000000$ (for $latex {i \in n}&fg=000000$ where $latex {n}&fg=000000$ is some index subset) is defined by the formula $latex {a \leq b \Leftrightarrow \forall i \in n : a_i \leq b_i}&fg=000000$. (By the way, it is a […]
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.