Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$.

I will denote the principal filters corresponding to a set $X$ as $\uparrow X$.

Let $a$ and $b$ are filters.

Suppose for some filter $M$ on binary relations we have $M\subseteq \uparrow A \times b$ and $M\subseteq a\times \uparrow B$ for some $A\in a$, $B\in b$.

Does it follow that there exist $X\in a$, $Y\in b$ such that $M\subseteq \uparrow X\times \uparrow Y$?

The above seems to be equivalent to the following conjecture, but formulated in elementary terms:

Conjecture $\left( \mathcal{A} \ltimes \mathcal{B} \right) \cup \left( \mathcal{A} \rtimes \mathcal{B} \right) = \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}$ for every filter objects $\mathcal{A}$, $\mathcal{B}$. (In words: join of oblique products is the reloidal product.)

See http://www.mathematics21.org/algebraic-general-topology.html for my research.

Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$.

I will denote the principal filter corresponding to a set $X$ as $\uparrow X$.

Let $a$ and $b$ are filters.

Suppose for some filter $M$ on binary relations we have $M\subseteq \uparrow A \times b$ and $M\subseteq a\times \uparrow B$ for some $A\in a$, $B\in b$.

Does it follow that there exist $X\in a$, $Y\in b$ such that $M\subseteq \uparrow X\times \uparrow Y$?

The above seems to be equivalent to the following conjecture, but formulated in elementary terms:

Conjecture $\left( \mathcal{A} \ltimes \mathcal{B} \right) \cup \left( \mathcal{A} \rtimes \mathcal{B} \right) = \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}$ for every filter objects $\mathcal{A}$, $\mathcal{B}$. (In words: join of oblique products is the reloidal product.)

See http://www.mathematics21.org/algebraic-general-topology.html for my research.