My first article is published

My first math article (titled “Filters on Posets and Generalizations”) was recently published in a peer reviewed, open access journal.

Why I published my first research article only in the age of 31? See my short autobiography.

Product of two funcoids and product of two funcoids

I’ve put online an article (PDF, a partial draft) where I define product of two morphisms for certain categories. (Such products are pointfree funcoids.) Particularly it is defined product of two funcoids and product of two reloids.

It is a more mature version of a draft I put online previously.

Some new theorems

I added the following theorems to Funcoids and Reloids article. The theorems are simple to prove but are surprising, as do something similar to inverting a binary relation which is generally neither monovalued nor injective.

Proposition Let , , are binary relations. Then .

Theorem Let , , are sets, , , . Then

Theorem Let , , are sets, , , . Then

The above theorems are the key for describing product funcoids, a task I previously got stuck. Now I can continue my research.

I am changing my research field

I failed to make progress in research of product of funcoids, the next thing I should research in my research plan. I also fail to solve any of my open problems.

Thus my research is stalled. I hope other people can solve the problems I formulated.

Due this crisis I decide to change my research field.

I am now going to research (instead of pure mathematics) semantics of XML namespaces and conversion between XML namespaces.

Micronization – the first attempt to define

This is my first attempt to define micronization.

Definition Let is a binary relation between sets and . micronization of is the complete funcoid defined by the formula (for every )

Conjecture If is a strict partial order, .

The idea of micronization is that it transforms a “global” relation (such as a strict partial order) into a “local” space (something like a topology).

This my definition probably can be generalized for funcoids instead of binary relations.

Product funcoids – a first messy draft

Product funcoids [outdated link remove] (not a math article but a messy collection of unproved and not exactly formulated statements).

This is my first attempt to define product funcoids. There is needed yet much work to rewrite it as a rigorous math text.

Orderings of filters in terms of reloids, draft

I updated the article Orderings of filters in terms of reloids from “preliminary draft” to just “draft”.

It means most errors are corrected and now you can read it.

The first problem in the chain is solved

I solved the first problem from this blog post (see Funcoids and Reloids article for a solution).

It opens the path for solving several other open problems which seem to be its consequences.

Path for solving my open problems

I will outline which open problems follow from other open problems. In this post I don’t enter into gory details how to prove these implications, because these are useless without a prior proof of the main premise. I write these notes just not to be forgotten.

It seems that from the first conjecture here follows this conjecture.

And from that conjecture follows the last conjecture (using the fact that ).

In turn the last conjecture may be used to prove properties of direct products of funcoids (more to write).

Two new conjectures

Conjecture If then for every funcoid and atomic f.o. and on the source and destination of correspondingly.

A stronger conjecture:

Conjecture If then for every funcoid and , .

Solution of these conjectures (specifically the first one) may help to prove other conjectures.