Funcoids and reloids in my math book draft
My partial draft book now contains the theory of funcoids and reloids. You may use this online draft (while I have not yet transferred the copyright) to study my theory of funcoids and reloids. Pointfree funcoids, multidimensional funcoids and staroids presented in my articles are not yet added to this partial draft.
A readable draft of “Multidimensional Funcoids”
I created a new draft of my article Multidimensional Funcoids article, which is probably has become readable now. Nevertheless there may be many errors yet. Now I am going to concentrate my efforts into putting my research in a book form for participating in EMS Monograph Award.
A change in terminology: multifuncoid -> staroid
I’ve made a change in terminology in my draft article Multidimensional funcoids: multifuncoid → staroid. I now use the term “multifuncoid” in an other sense. I made the change of the terminology in order for the meaning of the term “multifuncoid” to become more similar to the meaning of the term “pointfree funcoid”.
A conjecture related with subatomic product
With subatomic products first mentioned here and described in this article are related the following conjecture (or being precise three conjectures): Conjecture For every funcoid $latex f: \prod A\rightarrow\prod B$ (where $latex A$ and $latex B$ are indexed families of sets) there exists a funcoid $latex \Pr^{\left( A \right)}_k f$ defined by the formula $latex […]
A new theorem about subatimic product
I recently discovered what I call subatomic product of funcoids. Today I proved a simple theorem about subatomic product: Theorem $latex \prod^{\left( A \right)}_{i \in n} \left( g_i \circ f_i \right) = \prod^{\left( A \right)} g \circ \prod^{\left( A \right)} f$ for indexed (by an index set $latex n$) families $latex f$ and $latex g$ […]
Subatomic products – a new kind of product of funcoids
I’ve discovered a new kind of product of funcoids, which I call subatomic product. Definition Let $latex f : A_0 \rightarrow A_1$ and $latex g : B_0 \rightarrow B_1$ are funcoids. Then $latex f \times^{\left( A \right)} g$ (subatomic product) is a funcoid $latex A_0 \times B_0 \rightarrow A_1 \times B_1$ such that for every […]
A conjecture about direct product of funcoids
I am attempting to define direct products in the category cont(mepfFcd) (the category of monovalued, entirely defined continuous pointfree funcoids), see this draft article for a definition of this category. A direct product of objects may possibly be defined as the cross-composition product (see this article). A candidate for product of morphisms $latex f_1:\mathfrak{A}\rightarrow\mathfrak{B}$ and […]
“Categories related with funcoids”, a new draft
I started to write a new article Categories related with funcoids. It is now a very preliminary partial draft.
Is every isomorphisms of the category of funcoids a discrete funcoid?
The following is an important question related with categories related with funcoids: Question Is every isomorphisms of the category of funcoids a discrete funcoid?
Error corrected
In my draft article Multifuncoids there was a serious error. I defined funcoidal product wrongly. Now a new version of the article (with corrected error) is online.