# Generalization of the concept of duality principle

### Y. B. Karasik, Thoughts Guiding Systems Corp., Ottawa, Canada. e-mail:karasik@sympatico.ca

Mathematics has duality prinicples stating equivalence of some dual entities. The dual entities are entities, which certain elements are interchanged.

For example, in Boolean algebra dual entities are equations where operators "AND" and "OR" are interchanged as well as constants 1 and 0. In projective geometry on the plane dual entities are propositions about incidence of points and lines, where words "point" and "line" are interchanged. In optimization theory dual entities are formaulas where symbols Etc.

Duality principles in mathematics state logical equivalence of certain dual entities.

But entities which components are interchanged exist not in mathematics only. For instance, chemistry has a lot of such entities:

• N14H4+ + N15H3 is dual to N15H4+ + N14H3 as they have N14 and N15 interchanged;
• Li7H + Li6 is dual to Li6H + Li7 as they have Li7 and Li6 interchanged;
• B11F3 + B10Cl3 is dual to B10F3 + B11Cl3 as they have B10 and B11 interchanged;
• C13O2-3 + C12O2 is dual to C12O2-3 + C13O2;
• ATP + XDP is dual to ADP + XTP as they have interchanged letters T and D. (ATP is Adenosine triphosphate and ADP is Adenosine diphosphate; X is any nucleotide.)
Do duality principles exist for such dual entities ? If so these entities have to be equivalent in a sense. Are they chemically equivalent? Could be as it is known for a fact that they are by themselves mutually tranformed into each other:
• N14H4+ + N15H3 N15H4+ + N14H3
• Li7H + Li6 Li6H + Li7
• B11F3 + B10Cl3 B10F3 + B11Cl3
• C13O2-3 + C12O2 C12O2-3 + C13O2;
• ATP + XDP ADP + XTP
Such spontaneous mutual transformations are called ion exchange reactions.

But even if these dual chemical compositions are not chemically equivalent, then why would not extend the meaning of duality priciple as follows:

Duality principle is a statement of either equivalence of dual entities or their ability to convert into each other.