## $$*$$-dualities.(English)Zbl 0815.49026

Summary: For a complete, totally ordered group $$A= (A,\leq, *)$$ [in the sense of G. Birkhoff: “Lattice theory” (1948; Zbl 0033.101), Chapter 14], let $$\overline A= (\overline A, \leq,\overline {*},\underline {*})$$, where $$\overline A= A\cup \{+\infty\}\cup \{-\infty\}$$, with $$\leq$$ and $$*$$ extended from $$A$$ to $$\leq$$ and, respectively, $$\overline {*}$$ and $$\underline {*}$$, on $$\overline A$$, in a natural way, and, for any set $$X$$, extend $$\leq$$, $$\overline {*}$$ and $$\underline {*}$$ from $$\overline A$$ to $$\overline A^ X$$, pointwise on $$X$$. For two sets $$X$$ and $$W$$, we introduce and study $$*$$-dualities $$M: f\in \overline A^ X\to f^ M\in \overline A^ W$$ (by the conditions $$(\inf_{i\in I} f_ i)^ M= \sup_{i\in I} f^ M_ i$$ and $$(f\overline {*} a)^ M= f^ M{\underline {*}}a^{-1}$$ for all $$\{f_ i\}_{i\in I}\subseteq \overline A^ X$$, $$f\in \overline A^ X$$, $$a\in A$$), which encompass various kinds of known conjugations and polarities. We give some applications to $$(A,\leq,*)= (R, \leq, +)$$, $$(A, \leq,*)= (R_ +\backslash\{0\}, \leq, \times)$$, and to duality in optimization.

### MSC:

 49N15 Duality theory (optimization) 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 90C26 Nonconvex programming, global optimization

Zbl 0033.101
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