  An orthogonality space is a set equipped with a symmetric, irreflexive
relation called orthogonality. Every orthogonality space has an associated
complete ortholattice, called the logic of the orthogonality space. To every
poset, we associate an orthogonality space consisting of proper quotients (that
means, nonsingleton closed intervals), equipped with a certain orthogonality
relation. We prove that a finite bounded poset is a lattice if and only if the
logic of its orthogonality space is an orthomodular lattice. We prove that that
a poset is a chain if and only if the logic of the associated orthogonality
space is a Boolean algebra.

Rings and Algebras