Some Conjectures About Generalized Limits

Continuing my research from general topology monograph Algebraic General Topology, the following new open problems arose:

I remind that I define generalized limit of arbitrary function. This limit is defined in terms of funcoids. As I show in the Book 3, Algebra, generalized limit is defined for generalized spaces, for example for reloids.

So, how do generalized limits in terms of funcoids relate to generalized limits in terms of reloids?

If we have a generalized limit in terms of funcoids, can we calculate the generalized limit and terms of reloids, and vice versa?

Recalling from the formula of generalized limit,

lim f = { ν ∘ fr | rG }.

I ask the questions:

  • If ν is a reloid, can we build a bijection mapping \nu\circ f|_{\mathcal{A}} to (\mathsf{FCD})\nu\circ f|_{\mathcal{A}} where f ranges over monovalued functions and \mathcal{A} over filters?
  • If ν is a funcoid, can we build a bijection mapping (\mathsf{RLD})_{\mathrm{in}}\nu\circ f|{\mathcal{A}} to \nu\circ f|{\mathcal{A}} where f ranges over monovalued functions and \mathcal{A} over filters?

If such bijections exists, how do we relate generalized limits in terms of funcoids with generalized limits in terms of reloids? What is a generalized limit in reloids at all?

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