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August 6, 2011 / porton

Oblique product of filters

Funcoids and reloids are my research in the field of general topology.

Let \mathcal{A} and \mathcal{B} are filters.

Earlier I introduced three kinds of products of filters:

  • funcoidal product: \mathcal{A}\times^{\mathsf{FCD}}\mathcal{B};
  • reloidal product: \mathcal{A}\times^{\mathsf{RLD}}\mathcal{B};
  • second product: \mathcal{A}\times^{\mathsf{RLD}}_F\mathcal{B};

The last two products are reloids while the first is a funcoid.

Funcoidal and reloidal products are elegant math objects, while the second product is a stepchild primary useful for making counter-examples.

Today I found yet more two kind of products of filters which I call oblique products. They are something middle between reloidal product and second product, being similar to second product in one variable and reloidal product in the other variable. These may be also useful for making counter-examples.

They are defined by the formulas:

\mathcal{A} \ltimes \mathcal{B} = \\ \bigcap \{ \uparrow^{\mathsf{RLD} ( \mathrm{Base} ( \mathcal{A} ) ; \mathrm{Base} ( \mathcal{B} ) )} f \hspace{1em} | \\ \hspace{1em} f \in \mathscr{P} ( \mathrm{Base} ( \mathcal{A} ) \times \mathrm{Base} ( \mathcal{B} ) ), \forall B \in \mathrm{up} \mathcal{B} : \uparrow^{\mathsf{FCD} ( \mathrm{Base} ( \mathcal{A} ) ; \mathrm{Base} ( \mathcal{B} ) )} f \supseteq \mathcal{A} \times^{\mathsf{FCD}} \uparrow^{\mathrm{Base}(\mathcal{B})}B \};

\mathcal{A} \rtimes \mathcal{B} = \\ \bigcap \{ \uparrow^{\mathsf{RLD} ( \mathrm{Base} ( \mathcal{A} ) ; \mathrm{Base} ( \mathcal{B} ) )} f \hspace{1em} | \\ \hspace{1em} f \in \mathscr{P} ( \mathrm{Base} ( \mathcal{A} ) \times \mathrm{Base} ( \mathcal{B} ) ), \forall A \in \mathrm{up} \mathcal{A} : \uparrow^{\mathsf{FCD} ( \mathrm{Base} ( \mathcal{A} ) ; \mathrm{Base} ( \mathcal{B} ) )} f \supseteq \uparrow^{\mathrm{Base}(\mathcal{A})}A \times^{\mathsf{FCD}} \mathcal{B} \}.

Quickly arose the following conjectures:

Conjecture (\mathcal{A} \ltimes \mathcal{B} ) \cap ( \mathcal{A} \rtimes \mathcal{B} ) = \mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{B}.

Conjecture (\mathcal{A} \ltimes \mathcal{B} ) \cup ( \mathcal{A} \rtimes \mathcal{B} ) = \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}.

An informally formulated problem in addition to the above conjectures: Can we define an operation which transforms every funcoid into a reloid in a way similar to transforming a funcoidal product into an oblique product?

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