A note on implication operators of quantum logic

Abstract

Qiu (Notes on automata theory based on quantum logic, Sci China Ser F-Inf Sci, 2007, 50(2)154–169) discovered that some basic issues in orthomodular lattice-valued automata rely on bi-implication operator satisfying following condition: $$ (a_{1}\leftrightarrow b_{1} ) \wedge (a_{2}\leftrightarrow b_{2} )\leq a_{1} \wedge a_{2} \leftrightarrow b_{1} \wedge b_{2} ,~~\forall a_{1},a_{2} ,b_{1},b_{2} \in L, $$ and discovered that bi-implication operator based on Sasaki arrow satisfies this condition if and only if the truth-value lattice L is indeed a Boolean algebra, then asked a question of whether the result is also applied to other four quantum implication operators. We show that the answer is yes, and discuss several other conditions.