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November 27, 2014 / porton

A new function which is a counter-example to a conjecture found

For this conjecture there was found a counter-example, see this online article.

The counter-example states that (\mathsf{RLD})_{\Gamma} f \sqsupset (\mathsf{RLD})_{\mathrm{in}} f \sqsupset (\mathsf{RLD})_{\mathrm{out}} f for funcoid f=(=)|_{\mathbb{R}}.

This way I discovered a new function (\mathsf{RLD})_{\Gamma} defined by the formula (\mathsf{RLD})_{\Gamma} f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\,f ; \mathrm{Dst}\,f)} f.

While (\mathsf{RLD})_{\mathrm{in}} f and (\mathsf{RLD})_{\mathrm{out}} f have many cryptomorphisms (that is different but equivalent definitions) and lots of useful usages, all is known about (\mathsf{RLD})_{\Gamma} f is its definition and the formula (\mathsf{RLD})_{\Gamma} f \sqsupset (\mathsf{RLD})_{\mathrm{in}} f. It is unclear whether (\mathsf{RLD})_{\Gamma} is useful for anything.

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