• Algebraic and State Space Properties of Quantum Logics

This text contributes to the theory of quantum logics (the algebraic theory of the orthomodular posets). 
Chapter 1 introduces basic notions of quantum theories and their link with quantum logics.
Chapter 2  rigorously recalls the basic notions of the up-to-date quantum logics. 
Chapter 3  studies the algebraic generation in the lattice QLs. As a main result, we see how the fact of being locally finite allows for an extension of states. Then a Cartesian product of locally finite lattice QLs is investigated in view of the permanence property.
In Chapter 4, we investigate the QLs that are endowed with a symmetric difference. Main results obtained are finding examples of an irregular compatibility, proving the extension of ℤ2-valued states and presenting a construction of an example with a small state space and a big degree of non-compatibility. 
In Chapter 5, we endow the Abbott algebras with a symmetric difference (a kind of a XOR operation). We find that the Abbott XOR algebras are categorically equivalent to the class of lattice QLs with a symmetric difference. Another result is a description of compatibility. Also, Boolean algebras are characterized among the XOR Abbott algebras and an appropriate definition of a state is formulated and applied. 
In Chapter 6, we ask whether each set-representable quantum logic can be made point-distinguishing. We answer this question in the positive by considering an appropriate equivalence relation and, alternatively, by relating the problem to the Stone representation technique. 
In Chapter 7, we summarize the results and comment on the matters studied.