Kurosch-Ore's theorem

The set of Kurosch-Ore ( English Kurosh-Ore theorem or KUROS-Ore theorem ) is one of the classic sets of mathematical area of the lattice theory . The sentence deals with a question about irreducible representations of elements of modular associations and goes back to two publications that were submitted by the Soviet mathematician Alexander Gennadjewitsch Kurosch (in 1935) and by the Norwegian mathematician Øystein Ore (in 1936). It is related to Steinitz 's exchange theorem known from linear algebra and closely related to the isomorphism theorem for modular lattices, on which the proof of Kurosch-Ore's theorem is essentially based.

Formulation of the sentence

The sentence can be summarized as follows:

In a modular structure, every non-abbreviated representation of an element consisting of irreducible components (if any) always has the same number of components.

The following applies in detail:

Are given a modular lattice and two natural numbers and and elements and have both representations ${\displaystyle (V,\vee ,\wedge )}$ ${\displaystyle n\geq 1}$ ${\displaystyle m\geq 1}$ ${\displaystyle v,x_{0},\,\dots ,x_{n-1},y_{0},\,\dots ,y_{m-1}\in V}$ ${\displaystyle v}$

${\displaystyle v=x_{0}\vee \,\dots \vee x_{n-1}=y_{0}\vee \,\dots \vee y_{m-1}}$,

where the elements involved and all are -irreducible and both representations are -irredundant, ${\displaystyle x_{i}}$ ${\displaystyle y_{j}}$ ${\displaystyle \vee }$ ${\displaystyle \vee }$

so is ${\displaystyle n=m}$

while, for every index an index with ${\displaystyle i\in \{0,\,\dots ,n-1\}}$ ${\displaystyle j(i)\in \{0,\,\dots ,m-1\}}$

${\displaystyle v=x_{0}\vee \,\dots \vee x_{i-1}\vee y_{j(i)}\vee x_{i+1}\vee \,\dots \vee x_{n-1}}$.

The associated dual sentence applies in the same way .

Related sentences

I.

There are other versions of Kurosch-Ore's theorem. For example, in the monograph Lattices and Ordered Algebraic Structures by Thomas Scott Blyth, the sentence is offered in a different formulation that is essentially equivalent to the above, which states the following:

In a modular lattice that fulfills the descending chain condition, all irredundant representations of an element consisting of irreducible components have the same number of components. ${\displaystyle \vee }$ ${\displaystyle \vee }$

As Blyth shows, the Kurosch-Ore theorem can be tightened further in this version if, instead of a modular one, it is even based on a distributive lattice :

In a distributive lattice with descending chain condition, every lattice element different from the zero element has one and only one irredundant representation consisting of irreducible components . ${\displaystyle \vee }$ ${\displaystyle \vee }$

The last sentence also appears in the monograph Introduction to the Association Theory by Hans Hermes and is referred to there by the author as the decomposition sentence .

II

In his monograph, Hermes does not mention Kurosch-Ore's theorem, but in connection with the isomorphism theorem for modular associations, he formulates another theorem there, which is similar to Kurosch-Ore's theorem and which Hermes calls the chain theorem. This chain sentence can be represented as follows:

Are in the modular bandage two elements and by a finite chain and being at the same time maximum in the by inclusion ordered collection of sets of all and linking chains, as well as any other is and linking chain length and fulfills terms of their thickness , the inequality . ${\displaystyle (V,\vee ,\wedge )}$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle K_{0}\subseteq (V,\leq )}$ ${\displaystyle K_{0}}$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle K}$ ${\displaystyle |K|\leq |K_{0}|}$

The chain sentence is - after Richard Dedekind - also referred to as the Dedekind chain sentence and applies in the same way in every (up or down) semi-modular association .

When proving the chain theorem, Hermes again resorts to another result, which he obtains as a consequence of the mentioned isomorphism theorem and which he calls the neighboring theorem. In terms of content, this sentence makes the statement that in a modular association and also in the associated dual association, the semi-modular law is always fulfilled for two different elements . ${\displaystyle (V,\vee ,\wedge )}$ ${\displaystyle (V,\wedge ,\vee )}$ ${\displaystyle v,w\in V}$

Explanations and Notes

• In a bandage is for an element a representation ( English representation ) an equation of the form or the shape of a natural number . The one calls it the components of the representation. The number is the number of components. If necessary, one speaks more precisely of a representation or representation . ${\displaystyle (V,\vee ,\wedge )}$ ${\displaystyle v\in V}$ ${\displaystyle v=v_{0}\vee \,\dots \vee v_{n-1}}$ ${\displaystyle v=v_{0}\wedge \,\dots \wedge v_{n-1}}$ ${\displaystyle n\geq 0}$ ${\displaystyle v_{i}}$ ${\displaystyle n+1}$ ${\displaystyle \vee }$ ${\displaystyle \wedge }$
• Refers to a representation or as Redundant For ( English join-redundant ) or as Redundant For ( English meet-redundant ) if and only if there is an index available with or with . Otherwise, such a representation is called -irredundant ( English join-irredundant ) or -irredundant ( English meet-irredundant ). If the context is clear, one simply says redundant or irredundant . A redundant representation is in this sense can be shortened , while a irredundant representation unverkürzbar is. ${\displaystyle v=v_{0}\vee \,\dots \vee v_{n-1}}$ ${\displaystyle v=v_{0}\wedge \,\dots \wedge v_{n-1}}$ ${\displaystyle \vee }$ ${\displaystyle \wedge }$ ${\displaystyle i\in \{0,\,\dots ,n-1\}}$ ${\displaystyle v=v_{0}\vee \,\dots \vee v_{i-1}\vee v_{i+1}\vee \,\dots \vee v_{n-1}}$ ${\displaystyle v=v_{0}\wedge \,\dots \wedge v_{i-1}\wedge v_{i+1}\wedge \,\dots \wedge v_{n-1}}$ ${\displaystyle \vee }$ ${\displaystyle \wedge }$
• An element is -irreduzibel or vereinigungsirreduzibel ( English join-irreducible ) if and only if for out always or follows. Accordingly, an element is -irreduzibel or durchschnittsirreduzibel ( English meet-irreducible ) if and only if for out always or follows. If the context is clear, one simply says irreducible . The above association-theoretical concept of irreducibility corresponds to the concept of irreducibility in ring theory . ${\displaystyle v\in V}$ ${\displaystyle \vee }$ ${\displaystyle v_{1},v_{2}\in V}$ ${\displaystyle v=v_{1}\vee v_{2}}$ ${\displaystyle v=v_{1}}$ ${\displaystyle v=v_{2}}$ ${\displaystyle v\in V}$ ${\displaystyle \wedge }$ ${\displaystyle v_{1},v_{2}\in V}$ ${\displaystyle v=v_{1}\wedge v_{2}}$ ${\displaystyle v=v_{1}}$ ${\displaystyle v=v_{2}}$
• Each association is at the same time a partially ordered set , the order relation of which is obtained from the two connections and , which in turn is regained by forming the infimum and the supremum in pairs . This means that all terms known from order theory can be used in associations , and not least the term chain . Here, one says then that there are two different elements and by a chain connected if respect to the induced ordering relation a smallest and a largest member has and these two with and match. ${\displaystyle (V,\vee ,\wedge )}$ ${\displaystyle (V,\leq )}$ ${\displaystyle \vee }$ ${\displaystyle \wedge }$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle {\leq }|_{K}}$ ${\displaystyle v}$ ${\displaystyle w}$
• A partially ordered set meets the descending chain condition ( English descending chain condition ), where each chain of the form finite number of steps is stationary . A chain of form consisting of an infinite number of different elements is then impossible. The its dual concept is that of the ascending chain condition ( English ascending chain condition ). ${\displaystyle (V,\leq )}$ ${\displaystyle v_{1}\geq v_{2}\geq \dots \geq v_{n}\geq \dots }$ ${\displaystyle v_{1}>v_{2}>\dots >v_{n}>\dots }$
• According to Lev Anatolyevich Skornjakov , the association of the subspaces of a linear space (with inclusion as the ordering relation) is the most important example of a modular association , while (in general) the association of all subgroups is a group ... not a modular association .
• In his theory of associations, Helmuth Gericke emphasizes the normal division association of a group (with inclusion as the ordering relation) as an important example of a modular association. He reproduces Kurosch-Ore's theorem - without mentioning Kurosch and Ore - under the heading The exchange rate in modular associations .

literature

• Garrett Birkhoff : Lattice Theory (=  American Mathematical Society Colloquium Publications . Volume XXV ). 3. Edition. American Mathematical Society , Providence, RI 1967 ( MR0227053 ).
• TS Blyth: Lattices and Ordered Algebraic Structures (=  Universitext ). Springer-Verlag London, Ltd. , London 2005 ( MR2126425 ).
• Hans Hermes : Introduction to Association Theory (=  The Basic Teachings of Mathematical Sciences . Volume 73 ). Second expanded edition. Springer Verlag , Berlin, Heidelberg, New York 1967 ( MR0220634 ).
• Helmuth Gericke: Theory of Associations (=  university pocket books. 38 / 38a). 2nd Edition. Bibliographisches Institut , Mannheim 1967 ( MR0219453 ).
• George Grätzer : General Lattice Theory . New appendices by the author with BA Davey, R. Freese, B. Ganter , M. Greferath, P. Jipsen, HA Priestley, H. Rose, ET Schmidt , SE Schmidt, F. Wehrung and R. Wille . 2nd Edition. Birkhäuser Verlag , Basel, Boston, Berlin 1998, ISBN 3-7643-5239-6 ( MR1670580 ).
• AG Kuroš : Average representations with irreducible components in rings and so-called dual groups . In: Mat. Sb. Band 42 , 1935, pp. 613-616 .
• Ralph N. McKenzie , George F. McNulty , Walter F. Taylor : Algebras, Lattices, Varieties. Volume I (=  The Wadsworth & Brooks / Cole Mathematics Series ). Wadsworth & Brooks / Cole Advanced Books & Software, Monterey, California 1987, ISBN 0-534-07651-3 ( MR0883644 ).
• O. Ore : On the foundation of abstract algebra. II . In: Ann. of Math. (2) . tape 37 , 1936, pp. 265-292 ( MR1503277 ).
• Egon Splendor : Algebra of Associations (=  university paperbacks . Volume 958 ). Ferdinand Schöningh, Paderborn, Munich, Vienna, Zurich 1980, ISBN 3-506-99236-8 ( MR0637885 ).
• LA Skornjakow : Elements of Association Theory (=  Scientific Pocket Books, Series Mathematics / Physics . Volume 130 ). Akademie Verlag , Berlin 1973, ISBN 3-7643-5239-6 .
• Gábor Szász : Introduction to Association Theory . Akadémiai Kiadó , Budapest 1962 ( MR0138567 ).

Individual evidence

1. ^ Garrett Birkhoff: Lattice Theory. 1967, p. 75 ff., P. 166 ff.
2. ↑ George Graetzer: General Lattice Theory. 1998, p. 212 ff.
3. ^ LA Skornjakow: Elements of the association theory. 1973, p. 133 ff.
4. ↑ ^{a b} Ralph N. McKenzie et al .: Algebras, Lattices, Varieties. Volume I. 1987, p. 60
5. ^ Gábor Szász: Introduction to Association Theory. 1962, p. 109 ff., P. 166 ff.
6. ↑ Birkhoff, op. Cit., Pp. 75-76, p. 166
7. ↑ Grätzer, op. Cit., Pp. 212-213
8. ↑ Skornjakow, op. Cit., Pp. 133-134
9. ↑ Szász, op.cit., P. 111
10. ^ TS Blyth: Lattices and Ordered Algebraic Structures 2005, p. 60
11. ↑ Blyth, op. Cit., Pp. 69-70
12. ↑ Hans Hermes: Introduction to the Association Theory. 1967, p. 113
13. ↑ ^{a b} Hermes, op. Cit., Pp. 70–73
14. ^ Egon splendor: Algebra of associations. 1980, p. 106
15. ↑ Helmuth Gericke: Theory of Associations. 1967, p. 68 ff.
16. ↑ Hermes, op.cit., P. 70
17. ↑ Skornjakow, op.cit., P. 114
18. ↑ Gericke, op.cit., P. 78
19. ↑ Gericke, op. Cit., Pp. 143-146
20. ↑ Gericke in this context refers to Steinitz's exchange rate as the exchange rate of GRASSMANN and STEINITZ (op. Cit., P. 144).