Convex sets in modules over semifields

1. Convexsets in modulesoversemifields Karl-Ernst Biebler Institute for Biometry and Medical Informatics Ernst-Moritz-Arndt-University Greifswald Greifswald, Germany Email: biebler@biometrie.uni-greifswald.de

2. Outline • Vectorlatticesandsemifields • Modules oversemifields • S-convexsets in modulesoversemifields • S-norm and S-convexity • S-normabilityandinnerproduct • S-lineartopological S-module and S-norm • Extension theorems in S-modules • References

3. 1. Vectorlattices • A completeBooleanalgebra is isomorphic tothe open-closedsubsetsoftheextremallydisconnectedStoneanrepresentationspace • Completevectorlatticewithoder unit • CompleteBoolean Algebra ofidempotents

4. 1. Vectorlattices • setofcontinuousextended real functionsdefined on withvaluesoronly on nowheredensesubsetsof • the bounded functions in form a Stoneanalgebra • representationof a completevectorlattice: embedding

5. 1. Vectorlattices • contains always • Ifembedding of coincideswith, theniscalledextendedvectorlattice. • maybeatomic, atomeless, finite • If finite, so is isomorphic

6. 1. Vectorlattices • traceof , • is calledtheweak inverse of when and . • isan extendedvectorlatticeiff eachelementofisweakinvertible. • vector latticeofboundedelementsisweakinvertibleiff.

7. 1. Semifields Antonovski/Boltjaski/Sarymsakov (1960, 1963) A commutative assoziative ring with iscalledsemifield, if 0. +, + , 2. 3. supMexists in Sforeachboundedfromabove 4. 5. , has a solution in

8. 1. Semifields • An extendedvectorlatticewiththesetofnonnegativeelementsandthesetof all positive elementsiscalleduniversal semifield. • A Stoneanalgebrais a semifield. • A F-ordered ring in the sense ofGhika (1950) is an universal semifield.

9. 1. Topologicalsemifields Antonovski/Boltjaski/Sarymsakov (1960, 1963) A commutative assoziative topological ring withiscalledtopologicalsemifield, if ABS1. + , ABS2. ABS3. supMexists inforeachbounded fromabove ABS4. ABS5. , has a solution in

10. 1. Topologicalsemifields ABS6. - Boolean algebra ofidempotents of with the relative topology, withand a zeroneighborhood. Thenexist in such that. ABS7. Eachzeroneighborhood in contains a saturatedzeroneighborhood, thatmeans: Forwithholds. ABS8.Letbe a zeroneighborhood in . Thenexists a zeroneighborhoodin with .

11. 2. Modules oversemifields An Abeliangroupiscalled-module, ifthereis a multiplication with , , .

12. 3. S-convex sets in S-modules Let be a -module. iscalledS-convex: , for ; with iscalledstrong S-convex: , for; with

13. 3. S-convex sets in S-modules Letbe strong -convex. Then is -convex. The inverse statementis not true! Example: -module ; algebraic operationscoordinatewisedefined, is-convex. For, , therelationholds. Consequently, A is not strong -convex.

14. 3. S-convex sets in S-modules Separation Theorem: Let and strong -convexproper subsetsof a -module . Thenthereexistdisjointstrong -convexsets and insuch that, and .

15. 3. S-convex sets in S-modules Let be a -module. iscalledS-absorbing: For each there is such that iscalledS-circled: For all andall with hold .

16. 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 • from follows • for all , • + .

17. 4. S-norm and S-convexity Theorem Let be an universal semifieldanda -module. On exists a -norm iff there exists with • is strong - convex, • is - absorbing, • is - circled, • For each there is with .

18. 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 .

19. 5. S-normabilityandinnerproduct Let be a -module. A map from into is called S-inner product, if • ; iff Theorem In a -normed -module exists an S-inner product generating the -norm iff the parallelogramm identity holds. Classical result: Jordan/v.Neumann 1935 for normed vector spaces

20. 6. S-lineartopologicalS-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.

21. 6. S-lineartopologicalS-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 neccesaryiff is a finite dimensional .(Tychonov topology means the product topology.) Classical result on normability: Kolmogorov 1934

22. 7. Extension theorems in S-modules • NamiokaandDay: A monotone linear functionaldefined on a subspace (fulfillingcertainconditions) of a preorderedvectorspacecanextendedtothewholespace. • The Hahn-Banachtheoremisequivalenttotheextensiontheoremfor monotone linear functionals. • Werestrictourselfsto monotone S-linear maps.

23. 7. Extension theorems in S-modules Theorem: • LetSbe a semifieldwith atomar Boolean algebraofidempotentelements, • a preordered S-module, • a submoduleofsatisfying (B1), • : a monotone S-functional. Thenthereis an extensionf oftothewhole. Definition of (B1): Foreachthereiswith .

24. 7. Extension theorems in S-modules Theorem (O.T. Alas, 1973): • LetSbe an universal semifieldwith atomar Boolean algebraofidempotentelements, • a preordered S-module, • a submoduleofsatisfying (BA), • : a monotone S-functional. Thenthereis an extensionf oftothewhole. Definition of (BA): Foreachthereiswith .

25. Thankyouforyourattention !

26. 8. References • Alas OT: Semifields and positive linear functionals. Math. Japon. 18 (1973), 133-35 • AntonovskijMJa, Boltjanski BG, Sarymsakov TA: TopologicalSemifields (in Russian). Tashkent 1960 • AntonovskijMJa, BoltjanskiBG, SarymsakovTA: Topological Boolean Algebras (in Russian). Tashkent 1963 • Biebler KE: Extension theoremsandmodulesoversemifields (in German). Analysis Mathematica 15 (1989), 75-104 • Ghika A: Asuprainelelorcomutativeordonate(in Romanian). Buletinstiinti c Acad. Rep. Pop. Romine 2 (1950), 509-19 A moredetailedbibliography will befound in a publicationwhichis in preparation.