On an arbitrary meet-semilattice \(\textbf{S}=(S,\wedge ,0)\) with 0 we define an orthogonality relation and investigate the lattice \({{\,\mathrm{{\textbf {Cl}}}\,}}({\textbf {S}})\) of all subsets of S closed under this orthogonality. We show that if \(\textbf{S}\) is atomic then \({{\,\mathrm{{\textbf {Cl}}}\,}}(\textbf{S})\) is a complete atomic Boolean algebra. If \(\textbf{S}\) is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if \(\textbf{S}\) is a complete pseudocomplemented lattice then \({{\,\mathrm{{\textbf {Cl}}}\,}}(\textbf{S})\) 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.
