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September 4, 2010 / porton

A counter-example against a distributivity law for funcoids

Example There exist funcoids {f} and {g} such that

\displaystyle  ( \mathsf{RLD})_{\mathrm{out}} (g \circ f) \neq ( \mathsf{RLD})_{\mathrm{out}} g \circ ( \mathsf{RLD})_{\mathrm{out}} f.

Proof: Take {f = {( =)} |_{\Omega}} and {g = \mho \times^{\mathsf{FCD}} \left\{ \alpha \right\}} for some {\alpha \in \mho}. Then {( \mathsf{RLD})_{\mathrm{out}} f = \emptyset} and thus {( \mathsf{RLD})_{\mathrm{out}} g \circ ( \mathsf{RLD})_{\mathrm{out}} f = \emptyset}. We have {g \circ f = \Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}}. Let’s prove {( \mathsf{RLD})_{\mathrm{out}} (\Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}) = \Omega \times^{\mathsf{RLD}} \left\{ \alpha \right\}}.

Really:
( \mathsf{RLD})_{\mathrm{out}} (\Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}) = \\ \bigcap {\nobreak}^{\mathsf{RLD}} \mathrm{up} (\Omega \times^{\mathsf{FCD}} \left\{ \alpha \right\}) = \\ \bigcap {\nobreak}^{\mathsf{RLD}} \left\{ K \times \left\{ \alpha \right\} \hspace{1em} | \hspace{1em} K \in \mathrm{up} \Omega \right\} = \\ \bigcap {\nobreak}^{\mathfrak{F}} \left\{ K \hspace{1em} | \hspace{1em} K \in \mathrm{up} \Omega \right\} \times^{\mathsf{RLD}} \left\{ \alpha \right\} = \\ \Omega \times^{\mathsf{RLD}} \left\{ \alpha \right\}.

Thus {( \mathsf{RLD})_{\mathrm{out}} (g \circ f) = \Omega \times^{\mathsf{RLD}} \left\{ \alpha \right\} \neq \emptyset}. \Box

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