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March 29, 2010 / porton

Erroneous lemma corrected

In “Funcoids and Reloids” online draft there was an erroneous lemma:

Lemma For every two sets S and T of binary relations and every set A
\bigcap {\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} T \Rightarrow \bigcap {\nobreak}^{\mathfrak{F}} \{ \langle F \rangle A | F \in S \} = \bigcap{\nobreak}^{\mathfrak{F}} \{ \left\langle G \right\rangle A | G \in T \}.

The above lemma is false. The below modified lemma is true:

Lemma For every two filter bases S and T of binary relations and every set A
\bigcap {\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} T \Rightarrow \bigcap {\nobreak}^{\mathfrak{F}} \{ \langle F \rangle A | F \in S \} = \bigcap{\nobreak}^{\mathfrak{F}} \{ \left\langle G \right\rangle A | G \in T \}.

After correcting the lemma I corrected also the proof of the theorem which relies on this lemma:

Theorem (\mathsf{FCD}) (g \circ f) = ((\mathsf{FCD} g)) \circ ((\mathsf{FCD}) f) for every reloids f and g.

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