Dualities Associated to Binary Operations

Continuing our papers [14]–[16] and [6]–[9], where we have given an axiomatic approach to generalized conjugation theory, we introduce and study dualities $\Delta\colon \overline{R}^X \to \overline{R}^W$ 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 $\Delta\colon \overline{A}^X \to \overline{A}^W$ in the sense of [8], where Ā is the canonical enlargement of a complete totally ordered group.