I 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.

Earlier I claimed that I proved the following theorem:

$(\mathcal{A}\ltimes\mathcal{B})\sqcap(\mathcal{A}\rtimes\mathcal{B})=\mathcal{A}\times_{F}^{\mathsf{RLD}}\mathcal{B}$ for every filters $\mathcal{A}$, $\mathcal{B}$ on sets.

(Here $\ltimes$ and $\rtimes$ 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.

The considerations below were with an error, see the comment.

Product order ${\prod \mathfrak{A}}$ of posets ${\mathfrak{A}_i}$ (for ${i \in n}$ where ${n}$ is some index subset) is defined by the formula ${a \leq b \Leftrightarrow \forall i \in n : a_i \leq b_i}$. (By the way, it is a product in the category ${\mathbf{Pos}}$ of posets.)

By definition the lambda-function ${\lambda x \in D : F (x) = \left\{ (x ; F (x)) \mid x \in D \right\}}$ for a form ${F}$ depended on variable ${x}$.

It is easy to show that for a product of distributive lattices with least elements we have lattice-theoretic difference ${a \setminus b = \lambda i \in n : a_i \setminus b_i}$ whenever every ${a_i \setminus b_i}$ is defined. Compare also ${a \sqcup b = \lambda i \in n : a_i \sqcup b_i}$ (where ${\sqcup}$ denotes supremum of two elements), whenever every ${a_i \sqcup b_i}$ is defined.

I conjecture that this equality can be generalized to a relatively wide class of functions ${F}$ of a finite number of elements: ${F (x_0, \ldots, x_k) = \lambda i \in n : F (x_{0, i}, \ldots, x_{k, i})}$.

I do not hold the claim of originality of this conjecture. Moreover, I ask you to notify me (porton@narod.ru) if you know a work where a similar theory was described.

Now (to formulate the conjecture precisely) it is required to lay some formalistic.

I call an an order logic a set of propositional axioms (without quantifiers) with variables ${A}$, ${B}$, ${C}$, … and two binary relations ${=}$ and ${\leq}$ including at least the following axioms:

• ${A = A}$; ${A = B \Rightarrow B = A}$; ${A = B \wedge B = C \Rightarrow A = C}$ (equality axioms);
• ${A \leq A}$; ${A \leq B \wedge B \leq C \Rightarrow A \leq C}$; ${A \leq B \wedge B \leq A \Rightarrow A = B}$ (partial order axioms).

By definition a partial axiomatic function for a given order logic is a partial function of a finite number of arguments which is unambigously defined by some set of additional propositional formulas (the definition). I mean that we have some finite set of propositional formulas ${P_0 (y, x_0, \ldots, x_{k_0})}$, ${P_1 (y, x_0, \ldots, x_{k_1})}$, ${P_2 (y, x_0, \ldots, x_{k_2})}$, … such that it can be proved that ${y}$ is unambiguously determined by ${x_0}$, …, ${x_{k_0}}$, ${x_0}$, …, ${x_{k_1}}$, ${x_0}$, …, ${x_{k_2}}$, …

For example, the function ${\sqcup}$ is defined by the formula:

• ${A \sqcup B \leq C \Leftrightarrow A \leq C \wedge B \leq C}$.

That this definition of ${\sqcup}$ is unambiguous is a well known fact. Note that is general this function of two arguments is partial (as not every order is a semilattice).

Distributive lattices, Heyting algebras, and boolean algebras can be defined as examples of order logics.

Similarly we can define as partial axiomatic functions lattice-theoretic difference ${A \setminus B}$, boolean lattice complement and even co-brouwerian pseudodifference ${A \setminus^{\ast} B}$.

Conjecture: For every partial axiomatic function we have ${F (x_0, \ldots, x_k) = \lambda i \in n : F (x_{0, i}, \ldots, x_{k, i})}$ whenever every ${F (x_{0, i}, \ldots, x_{k, i})}$ is defined (Is it equivalent to ${F (x_0, \ldots, x_k)}$ to be defined?).

Please notify me (porton@narod.ru) if you know a solution of this conjecture.

Further generalization may be interesting. For example, our above consideration does not consider the formula ${\sup x = \lambda i \in n : \sup x_i}$ (where ${\sup}$ is the supremum on our poset), because supremum is a function of infinite arity, while we considered only finite relations.

Can it also be generalized for categorical product (not only in category ${\mathbf{Pos}}$)?

I have a little generalized the following old theorem:

$(a\sqcap^{\mathfrak{A}}b)^{\ast}=(a\sqcap^{\mathfrak{A}}b)^{+}=a^{\ast}\sqcup^{\mathfrak{A}}b^{\ast}=a^{+}\sqcup^{\mathfrak{A}}b^{+}$.

I have also found a new (easy to prove) theorem:

$(a\sqcup^{\mathfrak{A}}b)^{\ast}=(a\sqcup^{\mathfrak{A}}b)^{+}=a^{\ast}\sqcap^{\mathfrak{A}}b^{\ast}=a^{+}\sqcap^{\mathfrak{A}}b^{+}$.

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.

In a short (4 pages) article I define pointfree binary relations, a generalization of binary relations which does not use “points” (elements).

In a certain special case (of endo-relations) pointfree binary relations are essentially the same as binary relations.

It seems promising to research filters on sets of pointfree relations, generalizing the notion of reloids (that is filters on cartesian products) from my my research book

We need a more abstract way to define reloids:

For example filters on a set $A\times B$ are isomorphic to triples $(A;B;f)$ where $f$ is a filter on $A\times B$, as well as filters of boolean reloids (that is pairs $(\alpha;\beta)$ of functions $\alpha\in (\mathscr{P}B)^{\mathscr{P}A}$, $\beta\in (\mathscr{P}B)^{\mathscr{P}A}$ such that $y\sqcap \alpha x\neq\bot \Leftrightarrow x\sqcap \beta y\neq\bot$ (for all $x\in\mathscr{P}A$ and $y\in\mathscr{P}B$).

I propose a way to encompass all ways to describe reloids as follows:

Let call a filtrator of pointfree reloid a pair of a filtrator and an associative operation on its core. Then call abstract reloids pointfree reloids isomorphic (both a filtrators and as semigroups) to reloids.

I am yet unsure that this structure encompasses all essential properties of reloids (just like as primary filtrators encompass all properties of filters on posets).

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 point $x$.

It is currently an open problem to rewrite this proof in pointfree style.