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\(\bigvee\)-dualities and \(\perp\)-dualities. (English) Zbl 0728.06007

Summary: We introduce and study dualities \(\Delta\) : \({\bar {\mathbb{R}}}^ X\to {\bar {\mathbb{R}}}^ W\) (i.e., mappings \(f\in {\bar {\mathbb{R}}}^ X\to f^{\Delta}\in {\bar {\mathbb{R}}}^ W\) such that \((\inf_{i\in I}f_ i)^{\Delta}=\sup_{i\in I}f_ i^{\Delta}\) for all \(\{f_ i\}_{i\in I}\subseteq {\bar {\mathbb{R}}}^ X\) and all index sets I), which satisfy the additional condition \((f\vee d)^{\Delta}=f^{\Delta}\wedge -d\) (f\(\in {\bar {\mathbb{R}}}^ X\), \(d\in {\bar {\mathbb{R}}})\), and their duals, which are characterized as those dualities \(\Delta^*: {\bar {\mathbb{R}}}^ W\to {\bar {\mathbb{R}}}^ X\) for which \((f\perp d)^{\Delta^*}=f^{\Delta^*}\top -d\) (f\(\in {\bar {\mathbb{R}}}^ X\), \(d\in {\bar {\mathbb{R}}})\), where \(\perp\) and \(\top\) are two new binary operations on \({\bar {\mathbb{R}}}\), which we introduce here. Furthermore, we give a characterization of those \(\Delta\) which are also conjugations. Some applications are also mentioned.

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B23 Complete lattices, completions
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