\(*\)-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.


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
Full Text: DOI


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