On a certain mapping on the set with orthogonality. (English) Zbl 0672.06006

Havrda, Jan

Čas. Pěstování Mat. 114, No. 2, 160-164 (1989).

Given a set with orthogonality, (\(\Omega\),\(\perp)\), a mapping T: exp\(\Omega\) \(\to \exp \Omega\) is defined as follows: \(T(\emptyset)=\emptyset\), and \(T(A):=\{y\in \Omega:\) \(y\perp x\) for some \(x\in A\}\). This mapping is used to characterize maximal independent subsets of \(\Omega\), as well as to formulate a sufficient condition for equicardinality of two independent subsets. Also an isomorphism of the center of a certain orthomodular lattice into the powerset of a given set is described using T.

Reviewer: J.Cirulis

MSC:

06C15	Complemented lattices, orthocomplemented lattices and posets
03G12	Quantum logic

Keywords:

set with orthogonality; equicardinality of two independent subsets; orthomodular lattice