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

TL;DR

This paper introduces an orthogonality relation in semilattices and pseudocomplemented lattices, showing how it induces complete Boolean algebras of closed subsets, with results depending on atomicity and completeness.

Contribution

It defines a new orthogonality relation in semilattices and lattices, linking structural properties to Boolean algebra formations of closed sets.

Findings

Cl(S) is a complete atomic Boolean algebra if S is atomic

Cl(S) becomes a complete Boolean algebra if S is a complete pseudocomplemented lattice

Similar results hold for pseudocomplemented posets under certain conditions

Abstract

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.