commutativity relation in an orthocomplemented lattice
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 .
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1.
is reflexive.
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2.
iff .
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3.
iff .
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4.
if or , then .
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5.
is said to orthogonally commute with if and . In this case, we write . The terminology comes from the following fact: iff there are , with:
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(a)
( is orthogonal to , or ),
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(b)
,
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(c)
,
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(d)
, and
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(e)
.
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(a)
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6.
is symmetric iff iff is an orthomodular lattice.
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7.
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).
References
- 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 |
---|---|
Canonical name | CommutativityRelationInAnOrthocomplementedLattice |
Date of creation | 2013-03-22 16:43:22 |
Last modified on | 2013-03-22 16:43:22 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 6 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06C15 |
Classification | msc 03G12 |
Defines | dually commute |
Defines | orthogonally commute |