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We study a straightforward generalization, to vector spaces, of a counterexample of Zoltán Boros in which \( X=\bigl \{\,x\in \mathbb {R}^{\,2}: \ \ \ x_{1}+\,x_{2}\le 0\,\bigr \}\,; \) \( \varphi \,(x)=x_{1}+x_{2}\,, \quad \qquad f\,(x)=x+(1, \,-1)\,; \) \( S\,(x)=\bigl \{\,y\in X: \ \ \ \ \varphi \,(x)\le \varphi \,(y)\bigr \} \) for all \(x\in X\) . This example has, in particular, been used to show that an implication stated in a maximality theorem, published by  Raúl Fierro in 2017, is not true without assuming the antisymmetry of the corresponding preorder. A true particular case of this theorem improves and supplements a former similar theorem of  Sehie Park from 2000, and has to be proved just after  Zorn’s lemma and a  maximality principle of  H. Brézis and F. Browder.

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Generalization of an Instructive Counterexample of Zoltán Boros on Maximal Elements and Fixed Points in Preordered Sets

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  • 发表日期：

    2026年

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摘要

We study a straightforward generalization, to vector spaces, of a counterexample of Zoltán Boros in which \( X=\bigl \{\,x\in \mathbb {R}^{\,2}: \ \ \ x_{1}+\,x_{2}\le 0\,\bigr \}\,; \) \( \varphi \,(x)=x_{1}+x_{2}\,, \quad \qquad f\,(x)=x+(1, \,-1)\,; \) \( S\,(x)=\bigl \{\,y\in X: \ \ \ \ \varphi \,(x)\le \varphi \,(y)\bigr \} \) for all \(x\in X\) . This example has, in particular, been used to show that an implication stated in a maximality theorem, published by  Raúl Fierro in 2017, is not true without assuming the antisymmetry of the corresponding preorder. A true particular case of this theorem improves and supplements a former similar theorem of  Sehie Park from 2000, and has to be proved just after  Zorn’s lemma and a  maximality principle of  H. Brézis and F. Browder.