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

Árpád Száz ()
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Árpád Száz: University of Debrecen, Institute of Mathematics

Chapter 28 in Convex and Variational Analysis with Applications, 2026, pp 681-705 from Springer

Abstract: Abstract We study a straightforward generalization, to vector spaces, of a counterexample of Zoltán Boros in which $$ X=\bigl \{\,x\in \mathbb {R}^{\,2}: \ \ \ x_{1}+\,x_{2}\le 0\,\bigr \}\,; $$ X = { x ∈ R 2 : x 1 + x 2 ≤ 0 } ; $$ \varphi \,(x)=x_{1}+x_{2}\,, \quad \qquad f\,(x)=x+(1, \,-1)\,; $$ φ ( x ) = x 1 + x 2 , f ( x ) = x + ( 1 , - 1 ) ; $$ S\,(x)=\bigl \{\,y\in X: \ \ \ \ \varphi \,(x)\le \varphi \,(y)\bigr \} $$ S ( x ) = { y ∈ X : φ ( x ) ≤ φ ( y ) } for all $$x\in X$$ x ∈ 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.

Date: 2026
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DOI: 10.1007/978-3-032-07860-5_28

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