Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets
arxiv(2024)
摘要
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.
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