On J. Czelakowski’s Contributions to Quantum Logic and the Foundation of Quantum Mechanics

This paper provides an overview of Janusz Czelakowski’s contributions to the theory of partial Boolean ( \(\sigma \) -)algebras, and, more in general, to the foundation of Quantum Mechanics. Particular attention is paid to the logic of partial Boolean \(\sigma \) -algebras, to characterizations of PBAs embeddable into Boolean ( \(\sigma \) -)algebras, and their representation as self-adjoint idempotent elements of partial commutative algebras with involution. Also, applications to the theory of orthomodular posets as well as Czelakowski’s theory of partial Boolean algebras in a broader sense will be discussed. Finally, further representation theorems and their importance for quantum logic will be outlined.