New theorem about relationships between funcoids and reloids
I have proved (the proof is currently available in this file) that $latex ((\mathsf{FCD}), (\mathsf{RLD})_{\mathrm{in}})$ are components of a pointfree funcoid between boolean lattices. See my book for definitions.
A new (but easy to prove) theorem in my research book: Theorem Let $latex \mu$ and $latex \nu$ be endomorphisms of some partially ordered dagger precategory and $latex f\in\mathrm{Hom}(\mathrm{Ob}\mu;\mathrm{Ob}\nu)$ be a monovalued, entirely defined morphism. Then $latex f\in\mathrm{C}(\mu;\nu)\Leftrightarrow f\in\mathrm{C}(\mu^{\dagger};\nu^{\dagger}).$
$T_4$-funcoids
I have added to my free ebook a definition of $latex T_4$-funcoids (generalizing $latex T_4$ topologies). A funcoid $latex f$ is $latex T_4$ iff $latex f \circ f^{- 1} \circ f \circ f^{- 1} \sqsubseteq f \circ f^{- 1}$. This can also be generalized for pointfree funcoids.
A new easy theorem about pointfree funcoids
I have added the following easy to prove theorem to my general topology research book: Theorem If $latex \mathfrak{A}$ and $latex \mathfrak{B}$ are bounded posets, then $latex \mathsf{pFCD}(\mathfrak{A}; \mathfrak{B})$ is bounded.
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.