Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets

On an arbitrary meet-semilattice S with 0 we define an orthogonality relation and investigate the lattice Cl(S) of all subsets of S closed under this orthogonality. We show that if S is atomic then Cl(S) is a complete atomic Boolean algebra. If S is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if S is a complete pseudocomplemented lattice then Cl(S) is a complete Boolean algebra. For pseudocomplemented posets a similar result holds if the subset of pseudocomplements forms a complete lattice satisfying a certain compatibility condition.