Chajda, I., Kolařík, M., & Länger, H. (2025). Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets. ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS, 42(3), 577–592. https://doi.org/10.1007/s11083-025-09696-y
On an arbitrary meet-semilattice S = (S, ∧, 0) 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.
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