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…
read moreI 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…
read moreI’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…
read moreI 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…
read moreI 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 $latex \mathfrak{A}$…
read moreI 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…
read moreThe 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…
read moreI 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.
read moreI 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.
read moreEarlier 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…
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