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).
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 […]
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.
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 […]
“Open maps between funcoids” rewritten
There were several errors in the section “Open maps” of my online book. I have rewritten this section and also moved the section below in the book text. However, the new proof of the theorem stating that composition of open maps between funcoids is an open map now uses a proof referring to a particular […]
Pointfree funcoids between join-semilattices conjecture
Today I’ve come up with the following easy to prove theorem (exercise!) for readers of my book: Theorem If there exists at least one pointfree funcoid from a poset $latex \mathfrak{A}$ to a poset $latex \mathfrak{B}$ then either both posets have least element or none of them. This provokes me to the following conjecture also: […]
About topological structures corresponding to partial order
Intuitively (not in the sense of comparing cardinalities, but in some other sense), the set of natural numbers is less than the set of whole numbers, which is less than the set of rational numbers, which is less than the set of real numbers, which is less than the set of complex numbers. First, we […]