new properties of sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/tacl... ·...
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![Page 1: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/1.jpg)
New properties of Sasaki projections
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara*
* Czech Technical University in Prague** School of Mathematics, Johannesburg
TACL 2015, Ischia (Italy)
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 2: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/2.jpg)
Outline
What is quantum logic?
What is an orthomodular lattice (OML)?
What are commuting elements and what are their properties?
Sasaki and his projection
Some “known” properties of the Sasaki projection
Our results
Conclusion
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 3: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/3.jpg)
Outline
What is quantum logic?
What is an orthomodular lattice (OML)?
What are commuting elements and what are their properties?
Sasaki and his projection
Some “known” properties of the Sasaki projection
Our results
Conclusion
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 4: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/4.jpg)
Outline
What is quantum logic?
What is an orthomodular lattice (OML)?
What are commuting elements and what are their properties?
Sasaki and his projection
Some “known” properties of the Sasaki projection
Our results
Conclusion
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 5: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/5.jpg)
Outline
What is quantum logic?
What is an orthomodular lattice (OML)?
What are commuting elements and what are their properties?
Sasaki and his projection
Some “known” properties of the Sasaki projection
Our results
Conclusion
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 6: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/6.jpg)
Outline
What is quantum logic?
What is an orthomodular lattice (OML)?
What are commuting elements and what are their properties?
Sasaki and his projection
Some “known” properties of the Sasaki projection
Our results
Conclusion
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 7: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/7.jpg)
Outline
What is quantum logic?
What is an orthomodular lattice (OML)?
What are commuting elements and what are their properties?
Sasaki and his projection
Some “known” properties of the Sasaki projection
Our results
Conclusion
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 8: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/8.jpg)
Outline
What is quantum logic?
What is an orthomodular lattice (OML)?
What are commuting elements and what are their properties?
Sasaki and his projection
Some “known” properties of the Sasaki projection
Our results
Conclusion
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 9: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/9.jpg)
What is quantum logic?
1936Garrett Birkhoff and John von Neumann: The logic ofquantum mechanics. Annals of Mathematics, 37 (1936),823–843.
The structure of the lattice of projection operators P(H) on aHilbert space H.
The lattice of closed subspaces of a separable infinitedimensional Hilbert space.
Quantum logic can be characterised as the logical structurebased on orthomodular lattices.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 10: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/10.jpg)
What is quantum logic?
1936Garrett Birkhoff and John von Neumann: The logic ofquantum mechanics. Annals of Mathematics, 37 (1936),823–843.
The structure of the lattice of projection operators P(H) on aHilbert space H.
The lattice of closed subspaces of a separable infinitedimensional Hilbert space.
Quantum logic can be characterised as the logical structurebased on orthomodular lattices.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 11: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/11.jpg)
What is quantum logic?
1936Garrett Birkhoff and John von Neumann: The logic ofquantum mechanics. Annals of Mathematics, 37 (1936),823–843.
The structure of the lattice of projection operators P(H) on aHilbert space H.
The lattice of closed subspaces of a separable infinitedimensional Hilbert space.
Quantum logic can be characterised as the logical structurebased on orthomodular lattices.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 12: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/12.jpg)
What is quantum logic?
1936Garrett Birkhoff and John von Neumann: The logic ofquantum mechanics. Annals of Mathematics, 37 (1936),823–843.
The structure of the lattice of projection operators P(H) on aHilbert space H.
The lattice of closed subspaces of a separable infinitedimensional Hilbert space.
Quantum logic can be characterised as the logical structurebased on orthomodular lattices.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 13: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/13.jpg)
What is an orthomodular lattice?Orthocomplementation
Definition
An orthocomplementation on a bounded partially ordered set P isa unary operation with the properties
x ′ is the lattice–theoretical complement of x :
x ∧ x ′ = 0x ∨ x ′ = 1
′ is order reversing: x 6 y ⇒ y ′ 6 x ′
′ is an involution: x ′′ = x
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 14: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/14.jpg)
What is an orthomodular lattice?Orthomodular Lattices
Definition
An orthomodular lattice L is a lattice with anorthocomplementation in which the orthomodular law
x 6 y ⇒ y = x ∨ (x ′ ∧ y)
holds.
Recall:The lattice of closed subspaces of a separable infinite dimensionalHilbert space P(H)
is not distributive
is modular in case H has a finite dimension,
x 6 z ⇒ x ∨ (y ∧ z) = (x ∨ y) ∧ z .
is orthomodular in case of an infinite dimension.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 15: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/15.jpg)
What is an orthomodular lattice?Orthomodular Lattices
Definition
An orthomodular lattice L is a lattice with anorthocomplementation in which the orthomodular law
x 6 y ⇒ y = x ∨ (x ′ ∧ y)
holds.
Recall:The lattice of closed subspaces of a separable infinite dimensionalHilbert space P(H)
is not distributive
is modular in case H has a finite dimension,
x 6 z ⇒ x ∨ (y ∧ z) = (x ∨ y) ∧ z .
is orthomodular in case of an infinite dimension.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 16: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/16.jpg)
What is an orthomodular lattice?Orthomodular Lattices
Definition
An orthomodular lattice L is a lattice with anorthocomplementation in which the orthomodular law
x 6 y ⇒ y = x ∨ (x ′ ∧ y)
holds.
Recall:The lattice of closed subspaces of a separable infinite dimensionalHilbert space P(H)
is not distributive
is modular in case H has a finite dimension,
x 6 z ⇒ x ∨ (y ∧ z) = (x ∨ y) ∧ z .
is orthomodular in case of an infinite dimension.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 17: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/17.jpg)
What is an orthomodular lattice?Orthomodular Lattices
Definition
An orthomodular lattice L is a lattice with anorthocomplementation in which the orthomodular law
x 6 y ⇒ y = x ∨ (x ′ ∧ y)
holds.
Recall:The lattice of closed subspaces of a separable infinite dimensionalHilbert space P(H)
is not distributive
is modular in case H has a finite dimension,
x 6 z ⇒ x ∨ (y ∧ z) = (x ∨ y) ∧ z .
is orthomodular in case of an infinite dimension.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 18: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/18.jpg)
What is an orthomodular lattice?Orthomodular Lattices
Definition
An orthomodular lattice L is a lattice with anorthocomplementation in which the orthomodular law
x 6 y ⇒ y = x ∨ (x ′ ∧ y)
holds.
Recall:The lattice of closed subspaces of a separable infinite dimensionalHilbert space P(H)
is not distributive
is modular in case H has a finite dimension,
x 6 z ⇒ x ∨ (y ∧ z) = (x ∨ y) ∧ z .
is orthomodular in case of an infinite dimension.Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 19: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/19.jpg)
What are commuting elements?
Definition
Let L be an OML and x , y ∈ L, we say x commutes with y andwrite x C y if
x = (x ∧ y) ∨ (x ∧ y ′)
this relation is reflexive and symmetric.
transitive if and only if L is a Boolean algebra, in this case anypair of elements commutes.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 20: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/20.jpg)
What are commuting elements?
Definition
Let L be an OML and x , y ∈ L, we say x commutes with y andwrite x C y if
x = (x ∧ y) ∨ (x ∧ y ′)
this relation is reflexive and symmetric.
transitive if and only if L is a Boolean algebra, in this case anypair of elements commutes.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 21: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/21.jpg)
What are commuting elements?
Definition
Let L be an OML and x , y ∈ L, we say x commutes with y andwrite x C y if
x = (x ∧ y) ∨ (x ∧ y ′)
this relation is reflexive and symmetric.
transitive if and only if L is a Boolean algebra, in this case anypair of elements commutes.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 22: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/22.jpg)
What are commuting elements?
Definition
Let L be an OML and x , y ∈ L, we say x commutes with y andwrite x C y if
x = (x ∧ y) ∨ (x ∧ y ′)
this relation is reflexive and symmetric.
transitive if and only if L is a Boolean algebra, in this case anypair of elements commutes.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 23: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/23.jpg)
Some properties of the commuting relation
Lemma
Let L be an ortholattice and x , y ∈ L. If either x 6 y or x 6 y ′
then x and y commute.
Lemma
x C y ⇔ x ∧ (x ′ ∨ y) = x ∧ y
Proposition
In any orthomodular lattice, if x commutes with y and with z,then x commutes with y ′, with y ∨ z and with y ∧ z, as well aswith any (ortho–)lattice polynomial in variables y and z.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 24: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/24.jpg)
Some properties of the commuting relation
Lemma
Let L be an ortholattice and x , y ∈ L. If either x 6 y or x 6 y ′
then x and y commute.
Lemma
x C y ⇔ x ∧ (x ′ ∨ y) = x ∧ y
Proposition
In any orthomodular lattice, if x commutes with y and with z,then x commutes with y ′, with y ∨ z and with y ∧ z, as well aswith any (ortho–)lattice polynomial in variables y and z.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 25: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/25.jpg)
Some properties of the commuting relation
Lemma
Let L be an ortholattice and x , y ∈ L. If either x 6 y or x 6 y ′
then x and y commute.
Lemma
x C y ⇔ x ∧ (x ′ ∨ y) = x ∧ y
Proposition
In any orthomodular lattice, if x commutes with y and with z,then x commutes with y ′, with y ∨ z and with y ∧ z, as well aswith any (ortho–)lattice polynomial in variables y and z.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 26: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/26.jpg)
Some properties of the commuting relation
Lemma
Let L be an ortholattice and x , y ∈ L. If either x 6 y or x 6 y ′
then x and y commute.
Lemma
x C y ⇔ x ∧ (x ′ ∨ y) = x ∧ y
Proposition
In any orthomodular lattice, if x commutes with y and with z,then x commutes with y ′, with y ∨ z and with y ∧ z, as well aswith any (ortho–)lattice polynomial in variables y and z.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 27: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/27.jpg)
Sasaki and his projection
Definition
Let p be an element of the orthomodular lattice L and let ϕp amapping
ϕp : L → La 7→ p ∧ (p′ ∨ a)
then ϕp is called the Sasaki Projection of a into [0, p] ⊆ L.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 28: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/28.jpg)
Sasaki and his projection
Definition
Let p be an element of the orthomodular lattice L and let ϕp amapping
ϕp : L → La 7→ p ∧ (p′ ∨ a)
then ϕp is called the Sasaki Projection of a into [0, p] ⊆ L.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 29: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/29.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 30: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/30.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 31: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/31.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 32: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/32.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 33: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/33.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 34: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/34.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 35: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/35.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 36: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/36.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 37: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/37.jpg)
Some “known” properties
L is an OML ⇔ ϕp(q) = q ∀q 6 p
p ∧ q 6 ϕp(q) 6 p
Let L be an OML and p C q then ϕp(q) = p ∧ q
If y 6 z then ϕp(y) 6 ϕp(z).
p C q ⇔ ϕp(q) 6 q
ϕq(p) = 0⇔ p ⊥ q ⇔ p 6 q′
ϕq(p) = q ⇔ p′ ∧ q = 0
ϕp ◦ ϕq = ϕq ◦ ϕp = ϕp∧q ⇔ ϕp(q) = p ∧ q ⇔ p C q
p 6 q ⇔ ϕp = ϕq ◦ ϕp = ϕp ◦ ϕq
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 38: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/38.jpg)
Sasaki projection as binary operation
We will consider the Sasaki projection as a binary operation(sometimes called the Sasaki map)
∗ : L× L → L(x , y) 7→ x ∗ y = x ∧ (x ′ ∨ y)
The Sasaki projection is neither commutative nor associative, itsatisfies
idempotence x ∗ x = xneutral element 1 ∗ x = x ∗ 1 = xabsorption element 0 ∗ x = x ∗ 0 = 0
Further x C (x ∗ y), but not y C (x ∗ y)
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 39: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/39.jpg)
Sasaki projection as binary operation
We will consider the Sasaki projection as a binary operation(sometimes called the Sasaki map)
∗ : L× L → L(x , y) 7→ x ∗ y = x ∧ (x ′ ∨ y)
The Sasaki projection is neither commutative nor associative, itsatisfies
idempotence x ∗ x = xneutral element 1 ∗ x = x ∗ 1 = xabsorption element 0 ∗ x = x ∗ 0 = 0
Further x C (x ∗ y), but not y C (x ∗ y)
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 40: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/40.jpg)
Sasaki projection as binary operation
We will consider the Sasaki projection as a binary operation(sometimes called the Sasaki map)
∗ : L× L → L(x , y) 7→ x ∗ y = x ∧ (x ′ ∨ y)
The Sasaki projection is neither commutative nor associative, itsatisfies
idempotence x ∗ x = xneutral element 1 ∗ x = x ∗ 1 = xabsorption element 0 ∗ x = x ∗ 0 = 0
Further x C (x ∗ y), but not y C (x ∗ y)
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 41: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/41.jpg)
Sasaki projection as binary operation
We will consider the Sasaki projection as a binary operation(sometimes called the Sasaki map)
∗ : L× L → L(x , y) 7→ x ∗ y = x ∧ (x ′ ∨ y)
The Sasaki projection is neither commutative nor associative, itsatisfies
idempotence x ∗ x = xneutral element 1 ∗ x = x ∗ 1 = xabsorption element 0 ∗ x = x ∗ 0 = 0
Further x C (x ∗ y)
, but not y C (x ∗ y)
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 42: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/42.jpg)
Sasaki projection as binary operation
We will consider the Sasaki projection as a binary operation(sometimes called the Sasaki map)
∗ : L× L → L(x , y) 7→ x ∗ y = x ∧ (x ′ ∨ y)
The Sasaki projection is neither commutative nor associative, itsatisfies
idempotence x ∗ x = xneutral element 1 ∗ x = x ∗ 1 = xabsorption element 0 ∗ x = x ∗ 0 = 0
Further x C (x ∗ y), but not y C (x ∗ y)
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 43: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/43.jpg)
Other tools
Computer program (Marek Hycko)http://www.mat.savba.sk/˜hycko/omlCounterexamples could all be found in the orthomodular lattice L22
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Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 44: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/44.jpg)
Other tools
Computer program (Marek Hycko)http://www.mat.savba.sk/˜hycko/oml
Counterexamples could all be found in the orthomodular lattice L22
s
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s s s s s s s s s s ss s s s s s s s s s s
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Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 45: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/45.jpg)
Other tools
Computer program (Marek Hycko)http://www.mat.savba.sk/˜hycko/omlCounterexamples could all be found in the orthomodular lattice L22
s
s
s s s s s s s s s s ss s s s s s s s s s s
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Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 46: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/46.jpg)
Our results
Lemma
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. It has the following properties:
If x 6 y then x ′ ∨ (y ∗ z) = x ′ ∨ z.
If x 6 z then x ′ ∨ (y ∗ z) = x ′ ∨ y.
If x 6 z then (x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y.
If y 6 z then x ∗ y 6 x ∗ z.
If y 6 z then (x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y.
If z 6 y then x ∗ (y ∗ z) = (x ∗ y) ∗ z = x ∗ z.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 47: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/47.jpg)
Our results
Lemma
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. It has the following properties:
If x 6 y then x ′ ∨ (y ∗ z) = x ′ ∨ z.
If x 6 z then x ′ ∨ (y ∗ z) = x ′ ∨ y.
If x 6 z then (x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y.
If y 6 z then x ∗ y 6 x ∗ z.
If y 6 z then (x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y.
If z 6 y then x ∗ (y ∗ z) = (x ∗ y) ∗ z = x ∗ z.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 48: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/48.jpg)
Our results
Theorem
Let L be an OML with x, y and z elements in L. If x C y then
x ∗ (y ∗ z) = (x ∗ y) ∗ z
Proof idea:Calculate both sides and use the properties
x C y ⇒ x ∗ y = x ∧ y
x 6 y ⇒ x C y
if x commutes with y and with z , then x commutes with anyorthomodular lattice polynomial in variables y and z .
Note: Neither x C z nor y C z are sufficient to fulfil theassociativity equation; this is easy to find counterexamples inL22.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 49: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/49.jpg)
Our results
Theorem
Let L be an OML with x, y and z elements in L. If x C y then
x ∗ (y ∗ z) = (x ∗ y) ∗ z
Proof idea:Calculate both sides and use the properties
x C y ⇒ x ∗ y = x ∧ y
x 6 y ⇒ x C y
if x commutes with y and with z , then x commutes with anyorthomodular lattice polynomial in variables y and z .
Note: Neither x C z nor y C z are sufficient to fulfil theassociativity equation; this is easy to find counterexamples inL22.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 50: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/50.jpg)
Our results
Theorem
Let L be an OML with x, y and z elements in L. If x C y then
x ∗ (y ∗ z) = (x ∗ y) ∗ z
Proof idea:Calculate both sides and use the properties
x C y ⇒ x ∗ y = x ∧ y
x 6 y ⇒ x C y
if x commutes with y and with z , then x commutes with anyorthomodular lattice polynomial in variables y and z .
Note: Neither x C z nor y C z are sufficient to fulfil theassociativity equation; this is easy to find counterexamples inL22.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 51: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/51.jpg)
Our results
Theorem
Let L be an OML with x, y and z elements in L. If x C y then
x ∗ (y ∗ z) = (x ∗ y) ∗ z
Proof idea:Calculate both sides and use the properties
x C y ⇒ x ∗ y = x ∧ y
x 6 y ⇒ x C y
if x commutes with y and with z , then x commutes with anyorthomodular lattice polynomial in variables y and z .
Note: Neither x C z nor y C z are sufficient to fulfil theassociativity equation; this is easy to find counterexamples inL22.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 52: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/52.jpg)
Our results
Theorem
Let L be an OML with x, y and z elements in L. If x C y then
x ∗ (y ∗ z) = (x ∗ y) ∗ z
Proof idea:Calculate both sides and use the properties
x C y ⇒ x ∗ y = x ∧ y
x 6 y ⇒ x C y
if x commutes with y and with z , then x commutes with anyorthomodular lattice polynomial in variables y and z .
Note: Neither x C z nor y C z are sufficient to fulfil theassociativity equation; this is easy to find counterexamples inL22.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 53: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/53.jpg)
Our results
Theorem
Let L be an OML with x, y and z elements in L. If x C y then
x ∗ (y ∗ z) = (x ∗ y) ∗ z
Proof idea:Calculate both sides and use the properties
x C y ⇒ x ∗ y = x ∧ y
x 6 y ⇒ x C y
if x commutes with y and with z , then x commutes with anyorthomodular lattice polynomial in variables y and z .
Note: Neither x C z nor y C z are sufficient to fulfil theassociativity equation; this is easy to find counterexamples inL22.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 54: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/54.jpg)
Our results
Theorem
Let L be an OML with x, y and z elements in L. If x C y then
x ∗ (y ∗ z) = (x ∗ y) ∗ z
Proof idea:Calculate both sides and use the properties
x C y ⇒ x ∗ y = x ∧ y
x 6 y ⇒ x C y
if x commutes with y and with z , then x commutes with anyorthomodular lattice polynomial in variables y and z .
Note: Neither x C z nor y C z are sufficient to fulfil theassociativity equation; this is easy to find counterexamples inL22.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 55: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/55.jpg)
Alternative algebras
Definition
An alternative algebra M is a non–empty set together with a binaryoperation, ∗ which need not to be associative, but it has to bealternative, this means that ∀x , y ∈ M the following equations hold
x ∗ (x ∗ y) = (x ∗ x) ∗ y left identity and(y ∗ x) ∗ x = y ∗ (x ∗ x) right identity
From the left and right identities the flexible identity follows
x ∗ (y ∗ x) = (x ∗ y) ∗ x
Theorem
The Sasaki projection fulfils all three identities of an alternativealgebra.
It fulfils also (x ∗ x ′) ∗ y = x ∗ (x ′ ∗ y) = 0
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 56: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/56.jpg)
Alternative algebras
Definition
An alternative algebra M is a non–empty set together with a binaryoperation, ∗ which need not to be associative, but it has to bealternative, this means that ∀x , y ∈ M the following equations hold
x ∗ (x ∗ y) = (x ∗ x) ∗ y left identity and(y ∗ x) ∗ x = y ∗ (x ∗ x) right identity
From the left and right identities the flexible identity follows
x ∗ (y ∗ x) = (x ∗ y) ∗ x
Theorem
The Sasaki projection fulfils all three identities of an alternativealgebra.
It fulfils also (x ∗ x ′) ∗ y = x ∗ (x ′ ∗ y) = 0
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 57: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/57.jpg)
Alternative algebras
Definition
An alternative algebra M is a non–empty set together with a binaryoperation, ∗ which need not to be associative, but it has to bealternative, this means that ∀x , y ∈ M the following equations hold
x ∗ (x ∗ y) = (x ∗ x) ∗ y left identity and(y ∗ x) ∗ x = y ∗ (x ∗ x) right identity
From the left and right identities the flexible identity follows
x ∗ (y ∗ x) = (x ∗ y) ∗ x
Theorem
The Sasaki projection fulfils all three identities of an alternativealgebra.
It fulfils also (x ∗ x ′) ∗ y = x ∗ (x ′ ∗ y) = 0
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
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Alternative algebras
Definition
An alternative algebra M is a non–empty set together with a binaryoperation, ∗ which need not to be associative, but it has to bealternative, this means that ∀x , y ∈ M the following equations hold
x ∗ (x ∗ y) = (x ∗ x) ∗ y left identity and(y ∗ x) ∗ x = y ∗ (x ∗ x) right identity
From the left and right identities the flexible identity follows
x ∗ (y ∗ x) = (x ∗ y) ∗ x
Theorem
The Sasaki projection fulfils all three identities of an alternativealgebra.
It fulfils also (x ∗ x ′) ∗ y = x ∗ (x ′ ∗ y) = 0
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 59: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/59.jpg)
Moufang–like identities
Theorem
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. Then
(x ∗ y ∗ x) ∗ z = (x ∗ y) ∗ (x ∗ z)(z ∗ (x ∗ y)
)∗ x = z ∗ (x ∗ y ∗ x)(
(x ∗ y) ∗ z)∗ x = (x ∗ y) ∗ (z ∗ x)
for any x , y , z ∈ L.
Proof idea:
The Sasaki projection fulfils the left, right and flexibleidentities.Further we use the properties (x ∗ y) C x andIf x 6 z or y 6 z then
(x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 60: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/60.jpg)
Moufang–like identities
Theorem
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. Then
(x ∗ y ∗ x) ∗ z = (x ∗ y) ∗ (x ∗ z)(z ∗ (x ∗ y)
)∗ x = z ∗ (x ∗ y ∗ x)(
(x ∗ y) ∗ z)∗ x = (x ∗ y) ∗ (z ∗ x)
for any x , y , z ∈ L.
Proof idea:
The Sasaki projection fulfils the left, right and flexibleidentities.Further we use the properties (x ∗ y) C x andIf x 6 z or y 6 z then
(x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 61: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/61.jpg)
Moufang–like identities
Theorem
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. Then
(x ∗ y ∗ x) ∗ z = (x ∗ y) ∗ (x ∗ z)(z ∗ (x ∗ y)
)∗ x = z ∗ (x ∗ y ∗ x)(
(x ∗ y) ∗ z)∗ x = (x ∗ y) ∗ (z ∗ x)
for any x , y , z ∈ L.
Proof idea:
The Sasaki projection fulfils the left, right and flexibleidentities.Further we use the properties (x ∗ y) C x andIf x 6 z or y 6 z then
(x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 62: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/62.jpg)
Moufang–like identities
Theorem
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. Then
(x ∗ y ∗ x) ∗ z = (x ∗ y) ∗ (x ∗ z)(z ∗ (x ∗ y)
)∗ x = z ∗ (x ∗ y ∗ x)(
(x ∗ y) ∗ z)∗ x = (x ∗ y) ∗ (z ∗ x)
for any x , y , z ∈ L.
Proof idea:
The Sasaki projection fulfils the left, right and flexibleidentities.
Further we use the properties (x ∗ y) C x andIf x 6 z or y 6 z then
(x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 63: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/63.jpg)
Moufang–like identities
Theorem
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. Then
(x ∗ y ∗ x) ∗ z = (x ∗ y) ∗ (x ∗ z)(z ∗ (x ∗ y)
)∗ x = z ∗ (x ∗ y ∗ x)(
(x ∗ y) ∗ z)∗ x = (x ∗ y) ∗ (z ∗ x)
for any x , y , z ∈ L.
Proof idea:
The Sasaki projection fulfils the left, right and flexibleidentities.Further we use the properties (x ∗ y) C x and
If x 6 z or y 6 z then
(x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 64: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/64.jpg)
Moufang–like identities
Theorem
Let L be an orthomodular lattice and let ∗ be the Sasakiprojection. Then
(x ∗ y ∗ x) ∗ z = (x ∗ y) ∗ (x ∗ z)(z ∗ (x ∗ y)
)∗ x = z ∗ (x ∗ y ∗ x)(
(x ∗ y) ∗ z)∗ x = (x ∗ y) ∗ (z ∗ x)
for any x , y , z ∈ L.
Proof idea:
The Sasaki projection fulfils the left, right and flexibleidentities.Further we use the properties (x ∗ y) C x andIf x 6 z or y 6 z then
(x ∗ y) ∗ z = x ∗ (y ∗ z) = x ∗ y
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 65: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/65.jpg)
Conclusions
Sasaki operations form a promising alternative to latticeoperations (join and meet).
Sasaki operations satisfy more equations than mostorthomodular operations and therefore they are the bestcandidates for developing alternative algebraic tools forcomputations in orthomodular lattices.
The potential of using Sasaki operations in the algebraicfoundations of orthomodular lattices is still not sufficientlyexhausted.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 66: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/66.jpg)
Conclusions
Sasaki operations form a promising alternative to latticeoperations (join and meet).
Sasaki operations satisfy more equations than mostorthomodular operations and therefore they are the bestcandidates for developing alternative algebraic tools forcomputations in orthomodular lattices.
The potential of using Sasaki operations in the algebraicfoundations of orthomodular lattices is still not sufficientlyexhausted.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 67: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/67.jpg)
Conclusions
Sasaki operations form a promising alternative to latticeoperations (join and meet).
Sasaki operations satisfy more equations than mostorthomodular operations and therefore they are the bestcandidates for developing alternative algebraic tools forcomputations in orthomodular lattices.
The potential of using Sasaki operations in the algebraicfoundations of orthomodular lattices is still not sufficientlyexhausted.
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
![Page 68: New properties of Sasaki projectionslogica.dmi.unisa.it/tacl/wp-content/uploads/2014/08/TACL... · New properties of Sasaki projections Jeannine Gabri els*, Stephen Gagola III** and](https://reader035.vdocuments.us/reader035/viewer/2022071610/6149c7c312c9616cbc68fc08/html5/thumbnails/68.jpg)
Thanks
Thank you for your attention!
Questions?
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
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Thanks
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Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections
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Thanks
Thank you for your attention!
Questions?
Jeannine Gabriels*, Stephen Gagola III** and Mirko Navara* New properties of Sasaki projections