An important conjecture about funcoids. Version 2
This conjecture appeared to be false. Now I propose an alternative conjecture: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is nonempty. […]
An important conjecture about funcoids
Just a few minutes ago I’ve formulated a new important conjecture about funcoids: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is […]
Join of transitive reloids (a conjecture in uniformity theory)
Conjecture Join of a set $latex S$ on the lattice of transitive reloids is the join (on the lattice of reloids) of all compositions of finite sequences of elements of $latex S$. It was expired by theorem 2.2 in “Hans Weber. On lattices of uniformities”. There is a similar conjecture for funcoids (instead of reloids). […]
Preservation of properties of funcoids and reloids by their relationships
I have added a new section “Properties preserved by relationships” to my math research book. This section considers (in the form of theorems and conjectures) whether properties (reflexivity, symmetry, transitivity) of funcoids and reloids are preserved an reflected by their relationships (functions $latex (\mathsf{FCD})$, $latex (\mathsf{RLD})_{\mathrm{in}}$, $latex (\mathsf{RLD})_{\mathrm{out}}$ which map between funcoids and reloids).
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.