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ço between conjugations and coupling functions is one-to-one. 51) where a/0 = +Do, a/±oo = 0 (a E W+ ), and x is the "lower multiplication" on R + 1205 1 , that is, the extension of the usual multiplication x by the convention -Foo x O = O x +co = 0. , the extension of the usual multiplication x by the convention -Hoc X 0 = 0 X -koo = +oo.

Then A (e) A(; (2) if z then Cl(z) e; (3) CA (e) z e and AC(z) sets, (clearly, if (E, = (2X, D), (F, = (2' , D), where X and W are two then (A, e) is a Galois connection if and only if for any sets G C X and P C W, A (G) = G° and O(P) = P± are "polars" of G and P, respectively). 5 Dualities 15 has shown that if E and F are two complete lattices, then already one of the two mappings A and 8, say, A determines the Galois connection; namely (A, 8) is a Galois connection if and only if for any index set / we have A(inf ei) = sup A(e) ({eib E i c E), 8(z) = inf [x c E I A(x) ( (Z E F).

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Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) by Ivan Singer


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