Abstract

Continuing our papers [14]–[16] and [6]–[9], where we have given an axiomatic approach to generalized conjugation theory, we introduce and study dualities Δ ⁣:R‾X→R‾W\Delta\colon \overline{R}^X \to \overline{R}^W associated to a binary operation * on R̄, where X and W are two arbitrary sets and R̄ = [−∞, +∞], which encompass, as particular cases, conjugations, ∨-dualities and ⊥-dualities in the sense of [14] and [7]. We show that this class of dualities can be extended so as to encompass also the *-dualities Δ ⁣:A‾X→A‾W\Delta\colon \overline{A}^X \to \overline{A}^W in the sense of [8], where Ā is the canonical enlargement of a complete totally ordered group.

Author information

Contact details are reproduced from the original publication and may be historical.

Juan-Enrique Martínez-Legaz

Universitat Autònoma de Barcelona, Departament d’Economia i d’Història Econòmica, 08193 Bellaterra (Barcelona), Spain.

Ivan Singer

Institute of Mathematics, P.O. Box 1-764, 70700 Bucharest, Romania.

Keywords

  • Generalized conjugation theory
  • dualities

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Suggested citation

J.-E. Martínez-Legaz, I. Singer. “Dualities Associated to Binary Operations.” Journal of Convex Analysis 2 (1995), No. 1&2, 185–209. https://doi.org/10.68381/jca02013

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Published by Heldermann Verlag, 1995. Rights now held by Banach Press LLC.