Convex sets in modules over semifields
Karl-Ernst Biebler
Institute for Biometry and Medical Informatics Ernst-Moritz-Arndt-University Greifswald
Greifswald, Germany Email: biebler@biometrie.uni-greifswald.de
Outline
1. Vector lattices and semifields 2. Modules over semifields
3. S-convex sets in modules over semifields 4. S-norm and S-convexity
5. S-normability and inner product
6. S-lineartopological S-module and S-norm 7. Extension theorems in S-modules
8. References
1. Vector lattices
• A complete BOOLEAN algebra is isomorphic to the open-closed subsets of the extremally disconnected STONEAN representation space
• Complete vector lattice with oder unit
• Complete BOOLEAN Algebra of idempotents
•
1. Vector lattices
• set of continuous extended real functions defined on with values or only on nowhere dense subsets of
• the bounded functions in form a STONEAN algebra
• representation of a complete vector lattice : embedding
•
1. Vector lattices
• contains always
• If embedding of coincides with , then is called extended vector lattice.
• may be atomic, atomeless, finite
• If finite, so is isomorphic
•
1. Vector lattices
• trace of ,
• is called the weak inverse of when and .
• is an extended vector lattice iff
each element of is weak invertible.
• vector lattice of bounded elements is weak invertible iff .
•
1. Semifields
ANTONOVSKI/BOLTJASKI/SARYMSAKOV (1960, 1963) A commutative assoziative ring with
is called semifield, if
0.
+ , + , 2.
3. sup M exists in S for each bounded from above 4.
5. , has a solution in
•
1. Semifields
• An extended vector lattice with the set of nonnegative elements and the set of all
positive elements is called universal semifield.
• A STONEAN algebra is a semifield.
• A F-ordered ring in the sense of GHIKA (1950) is an universal semifield.
•
1. Topological semifields
ANTONOVSKI/BOLTJASKI/SARYMSAKOV (1960, 1963)
A commutative assoziative topological ring with is called topological semifield, if
ABS1. + , ABS2.
ABS3. sup M exists in for each bounded from above
ABS4.
ABS5. , has a solution in
•
1. Topological semifields
ABS6. - BOOLEAN algebra of idempotents of with the relative topology, with and a zero neighborhood. Then
exist in such that .
ABS7. Each zero neighborhood in contains a saturated zero neighborhood , that means:
For with holds .
ABS8. Let be a zero neighborhood in . Then exists a zero neighborhood in with
.
•
2. Modules over semifields
An ABELIAN group is called -module, if there is a multiplication with
1.
2.
3.
4.
, , .
•
3. S-convex sets in S-modules
Let be a -module.
is called S-convex: , for ; with
is called strong S-convex: , for ; with
•
3. S-convex sets in S-modules
Let be strong -convex. Then is -convex.
The inverse statement is not true!
Example:
-module ; algebraic operations coordinatewise defined, is -convex. For , , the relation holds.
Consequently, A is not strong -convex.
•
3. S-convex sets in S-modules
Separation Theorem:
Let and strong -convex proper subsets of a -module . Then there exist disjoint strong -convex sets and in such that , and .
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3. S-convex sets in S-modules
Let be a -module.
is called S-absorbing:
For each there is such that is called S-circled:
For all and all with hold .
•
4. S-norm and S-convexity
A norm can be defined on every real vector lattice.
Let be a -module. is called -normed -module, if there exists a map from into with
1. from follows 2. for all ,
3. + .
•
4. S-norm and S-convexity
Theorem
Let be an universal semifield and a -module. On exists a -norm iff there exists with
1. is strong - convex, 2. is - absorbing,
3. is - circled,
4. For each there is with .
•
4. S-norm and S-convexity
Is the Theorem valid for arbitrary semifields?
OPEN ! Remark
In a -module there is no analogue to a linear base in a real vector lattice.
Corollary
Let be an universal semifield and a free -module.
Then a -norm exists on .
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5. S-normability and inner product
Let be a -module. A map from into is called S-inner product, if
1. ; iff 2.
3.
4.
Theorem
In a -normed -module exists an S-inner product generating the -norm iff the parallelogramm identity
holds.
•
6. S-lineartopological S-module and S-norm
topological semifield, a - module. is called
-lineartopological -module, if is a HAUSDORFF topology suitable to the algebraic structure.
Theorem
Let be a -normed -module and the na- tural
topology on . Zero neighborhood base for are sets and runs through a zero neighborhood base of .
Then is a -lineartopological -module.
•
6. S-lineartopological S-module and S-norm
Theorem
Let be an universal topological semifield and a -lineartopological -module.
The existence of a -bounded and strong -convex zero neighborhood in is sufficient for the
S-normability of .
It is neccesary iff is a finite dimensional .(TYCHONOV topology means the product topology.)
Classical result on normability: KOLMOGOROV 1934
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7. Extension theorems in S-modules
• NAMIOKA and DAY: A monotone linear functional defined on a subspace (fulfilling certain conditions) of a preordered vector space can extended to the whole space.
• The HAHN-BANACH theorem is equivalent to the extension theorem for monotone linear functionals.
• We restrict ourselfs to monotone S-linear maps.
7. Extension theorems in S-modules
Theorem:
• Let S be a semifield with atomar Boolean algebra of idempotent elements,
• a preordered S-module,
• a submodule of satisfying (B1),
• : a monotone S-functional.
Then there is an extension f of to the whole .
Definition of (B1):
For each there is with .
•
7. Extension theorems in S-modules
Theorem (O.T. ALAS, 1973):
• Let S be an universal semifield with atomar Boolean algebra of idempotent elements,
• a preordered S-module,
• a submodule of satisfying (BA),
• : a monotone S-functional.
Then there is an extension f of to the whole . Definition of (BA):
For each there is with .
•
Thank you for your attention !
8. References
• Alas OT: Semifields and positive linear functionals. Math. Japon. 18 (1973), 133-35
• Antonovskij MJa, Boltjanski BG, Sarymsakov TA: Topological Semifields (in Russian).
Tashkent 1960
• Antonovskij MJa, Boltjanski BG, Sarymsakov TA: Topological Boolean Algebras (in Russian). Tashkent 1963
• Biebler KE: Extension theorems and modules over semifields (in German).
Analysis Mathematica 15 (1989), 75-104
• Ghika A: Asupra inelelor comutative ordonate (in Romanian).
Buletin stiinti c Acad. Rep. Pop. Romine 2 (1950), 509-19
A more detailed bibliography will be found in a publication which is in preparation.
