**Mladen Pavicic ^{1,2} and
Norman D. Megill^{3}
**

^{1}*Atominstitute of the Austrian Universities,
Schüttelstraße 115, A-1020 Wien, Austria; pavicic@ati.ac.at
*

^{2}*
Department of Mathematics, University of Zagreb,
GF, Kaciceva 26, POB-217, HR-10001 Zagreb, Croatia;
mpavicic@faust.irb.hr; http://m3k.grad.hr/pavicic
*

^{3}*
Locke Lane, Lexington, MA 02173, U. S. A.; nm@alum.mit.edu
*

** Abstract. **
Join in an orthomodular lattice is obtained in the same form
for all five quantum implications. The form holds for the classical
implication in a distributive lattice as well. Even more, the definition
added to an ortholattice makes it orthomodular for quantum implications
and distributive for the classical one. Based on this result a quantum
implication algebra with a single primitive - and in this sense
unique - implication is formulated. A corresponding classical
implication algebra is also formulated. The algebras are shown to be
special cases of a universal implication algebra.

**PACS numbers: ** 03.65.Bz, 02.10.By, 02.10.Gd

**Keywords: ** implication algebra, orthomodular
lattices, quantum logic, classical logic.