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\(*\)-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

Citations:

Zbl 0033.101
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References:

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