Abstract
Abstract On an arbitrary meet-semilattice $$\textbf{S}=(S,\wedge ,0)$$ S = ( S , ∧ , 0 ) with 0 we define an orthogonality relation and investigate the lattice $${{\,\mathrm{{\textbf {Cl}}}\,}}({\textbf {S}})$$ Cl ( S ) of all subsets of S closed under this orthogonality. We show that if $$\textbf{S}$$ S is atomic then $${{\,\mathrm{{\textbf {Cl}}}\,}}(\textbf{S})$$ Cl ( S ) is a complete atomic Boolean algebra. If $$\textbf{S}$$ S is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if $$\textbf{S}$$ S is a complete pseudocomplemented lattice then $${{\,\mathrm{{\textbf {Cl}}}\,}}(\textbf{S})$$ 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.
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call