An equational theory for σ -complete orthomodular lattices

The condition of σ-completeness related to orthomodular lattices places an important role in the study of quantum probability theory. In the framework of algebras with infinitary operations, an equational theory for the category of σ-complete orthomodular lattices is given. In this structure, we study the congruences theory and directly irreducible algebras establishing an equational completeness theorem. Finally, a Hilbert style calculus related to σ-complete orthomodular lattices is introduced and a completeness theorem is obtained