commutativity relation in an orthocomplemented lattice
Some properties. Below are some properties of the commutativity relations and .
if or , then .
is said to orthogonally commute with if and . In this case, we write . The terminology comes from the following fact: iff there are , with:
( is orthogonal to , or ),
is symmetric iff iff is an orthomodular lattice.
Remark. More generally, one can define commutativity on an orthomodular poset : for , iff , , and exist, and . Dual commutativity and mutual commutativity in an orthomodular poset are defined similarly (with the provision that the binary operations on the pair of elements are meaningful).
- 1 L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
|Title||commutativity relation in an orthocomplemented lattice|
|Date of creation||2013-03-22 16:43:22|
|Last modified on||2013-03-22 16:43:22|
|Last modified by||CWoo (3771)|