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commutativity relation in an orthocomplemented lattice
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(Definition)
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Let be an orthocomplemented lattice with . We say that commutes with if
. When commutes with , we write
. Dualize everything, we have that dually commutes with , written
, if
.
Some properties. Below are some properties of the commutativity relations
and
.
-
is reflexive.
-
iff
.
-
iff
.
- 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
),
,
,
, and
.
-
is symmetric iff
iff is an orthomodular lattice.
-
is an equivalence relation iff
iff is a Boolean algebra.
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).
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"commutativity relation in an orthocomplemented lattice" is owned by CWoo.
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| Also defines: |
dually commute, orthogonally commute |
This object's parent.
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Cross-references: binary operations, orthomodular poset, Boolean algebra, equivalence relation, orthomodular lattice, symmetric, orthogonal, iff, Reflexive, relations, commutativity, properties, orthocomplemented lattice
There is 1 reference to this entry.
This is version 3 of commutativity relation in an orthocomplemented lattice, born on 2007-02-21, modified 2007-03-10.
Object id is 8943, canonical name is CommutativityRelationInAnOrthocomplementedLattice.
Accessed 1216 times total.
Classification:
| AMS MSC: | 06C15 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented lattices, orthocomplemented lattices and posets) | | | 03G12 (Mathematical logic and foundations :: Algebraic logic :: Quantum logic) |
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Pending Errata and Addenda
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