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March 17, 2011 / porton

Two similar theorems about funcoids and reloids

I proved the following two similar theorems about funcoids and reloids:

Theorem

  1. For a complete funcoid f there exist exactly one function
    F \in \mathfrak{F}^{\mho} such that
    f = \bigcup^{\mathsf{FCD}} \left\{ \{ \alpha \} \times^{\mathsf{FCD}} F(\alpha) | \alpha \in \mho \right\}.
  2. For a co-complete funcoid f there exist exactly one function
    F \in \mathfrak{F}^{\mho} such that
    f = \bigcup^{\mathsf{FCD}} \left\{ F(\alpha) \times^{\mathsf{FCD}} \{ \alpha \} | \alpha \in \mho \right\}.

Theorem

  1. For a complete reloid f there exist exactly one function
    F \in \mathfrak{F}^{\mho} such that
    f = \bigcup^{\mathsf{RLD}} \left\{ \{ \alpha \} \times^{\mathsf{RLD}} F(\alpha) | \alpha \in \mho \right\}.
  2. For a co-complete reloid f there exist exactly one function
    F \in \mathfrak{F}^{\mho} such that
    f = \bigcup^{\mathsf{RLD}} \left\{ F(\alpha) \times^{\mathsf{RLD}} \{ \alpha \} | \alpha \in \mho \right\}.

See this online article for definitions of used concepts and proofs.

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