A new theorem proved
I have proved $latex (\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}} = \Omega^{\mathsf{RLD}}$ (where $latex \Omega^{\mathsf{FCD}}$ is a cofinite funcoid and $latex \Omega^{\mathsf{RLD}}$ is a cofinite reloid that is reloid defined by a cofinite filter). The proof is currently available in this draft. Note that in the previous draft there was a wrong formula for $latex (\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}}$.
A typo in my math book
I’ve found a typo in my math book. I confused existential quantifiers with universal quantifiers in the section “Second product. Oblique product” in the chapter “Counter-examples about funcoids and reloids”.
More on generalized cofinite filters
I added more properties of cofinite funcoids to this draft.
Generalized cofinite filters
I have described generalized cofinite filters (including the “cofinite funcoid”). See the draft at http://www.math.portonvictor.org/binaries/addons.pdf
A new diagram about funcoids and reloids
Define for posets with order $latex \sqsubseteq$: $latex \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$; $latex \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \}$. Note that the above is a generalization of monotone Galois […]
A new kind of product of funcoids
The following is one of a few (possibly non-equivalent) definitions of products of funcoids: Definition Let $latex f$ be an indexed family of funcoids. Let $latex \mathcal{F}$ be a filter on $latex \mathrm{dom}\, f$. $latex a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b \Leftrightarrow \exists N \in \mathcal{F} \forall i \in N : \mathrm{Pr}^{\mathsf{RLD}}_i\, a \mathrel{[f_i]} \mathrm{Pr}^{\mathsf{RLD}}_i\, […]
A different definition of product of funcoids
Definition $latex a \mathrel{\left[ \prod^{(A 2)} f \right]} b \Leftrightarrow \exists M \in \mathrm{fin} \forall i \in (\mathrm{dom}\, f) \setminus M : \Pr^{\mathsf{RLD}}_i a \mathrel{[f_i]} \Pr^{\mathsf{RLD}}_i b$ for an indexed family $latex f$ of funcoids and atomic reloids $latex a$ and $latex b$ of suitable form. Here $latex M \in \mathrm{fin}$ means that $latex M$ […]
A conjecture proved
I’ve found an easy positive proof of this my conjecture.
Partial order and micronization (conjecture)
Conjecture $latex S^{\ast}(\mu(E)) = E$ for every partial order $latex E$. The function $latex S^\ast$ and micronization $latex \mu$ are defined in my research monograph.
A new little theorem (Galois connections)
I’ve added the following to my research book: Definition Galois surjection is the special case of Galois connection such that $latex f^{\ast} \circ f_{\ast} $ is identity. Proposition For Galois surjection $latex \mathfrak{A} \rightarrow \mathfrak{B}$ such that $latex \mathfrak{A}$ is a join-semilattice we have (for every $latex y \in \mathfrak{B}$) $latex f_{\ast} y = \max […]