Lemma of Iwamura

The Iwamura lemma is a doctrine of the mathematical field of order theory and goes back to a scientific publication by the mathematician Tsurane Iwamura in 1944. The lemma deals with a question about the coverage representability of directed sets and gave rise to a series of further investigations.

Formulation of the lemma

Following on from Marcel Erné's monograph Introduction to Order Theory , the lemma can be presented as follows:

For every infinite partially ordered set , which is directed upwards by the underlying order relation , there is a subset system with the following properties: ${\ displaystyle (M, \ leq)}$ ${\ displaystyle \ leq}$ ${\ displaystyle {\ mathcal {K}} \ subset 2 ^ {M}}$

(I) is a chain of sets well ordered by the inclusion relation . ${\ displaystyle {\ mathcal {K}}}$

(II) The subsets cover the basic set ; so it applies . ${\ displaystyle K \ in {\ mathcal {K}}}$ ${\ displaystyle M}$ ${\ displaystyle \ bigcup {\ mathcal {K}} = M}$

(III) For each subset is considered in terms of their thickness , the inequality . ${\ displaystyle K \ in {\ mathcal {K}}}$ ${\ displaystyle | K | <| M |}$

(IV) For each subset , the partially ordered set provided with the induced order relation is also directed upwards. ${\ displaystyle K \ in {\ mathcal {K}}}$ ${\ displaystyle (K, {\ leq} | _ {K})}$

Explanations and Notes

The lemma can be traced back to the fact that every infinite set can be represented as a union of a subset system of , which is well ordered by the inclusion relation , in which each of the subsets within it has a really smaller thickness than the set itself. ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle M}$

The lemma is based on the fact that for an infinite set the subset system of finite subsets and the set itself are always of equal power . ${\ displaystyle M}$ ${\ displaystyle M}$

As an application one can show by means of the lemma of Iwamura that a partially ordered set is already complete (i.e. every upwardly directed subset has a smallest upper bound) if every upwardly directed chain has a smallest upper bound.

literature

Marcel Erné: Introduction to Order Theory . Bibliographisches Institut , Mannheim 1982, ISBN 3-411-01638-8 ( MR0689891 ).
T. Iwamura: A Proposition About Directed Sets (Japanese) . In: Zenkoku Shijo Sugaku Danwakai . tape 262 , 1944, pp. 107-111 .
George Grätzer : General Lattice Theory (= textbooks and monographs from the field of exact sciences, mathematical series . Volume 52 ). Birkhäuser Verlag , Basel, Stuttgart 1978, ISBN 3-7643-0813-3 ( MR0504338 ).
George Markowsky: Chain-complete posets and directed sets with applications . In: Algebra Universalis . tape 6 , 1976, p. 53-68 , doi : 10.1007 / BF02485815 ( MR0398913 ).
J. Mayer-Kalkschmidt, E. Steiner: Some theorems in set theory and applications in the ideal theory of partially ordered sets . In: Duke Mathematical Journal . tape 31 , 1964, pp. 287-289 ( MR0160729 ).

Individual evidence

↑ ^a ^b ^c Marcel Erné: Introduction to the theory of order. 1982, p. 230
↑ George Graetzer: General Lattice Theory. 1978, p. 122
↑ Erné, op.cit., P. 229
↑ This can be proven with the help of Zorn's lemma .
↑ Steven Roman: Lattices and Ordered Sets , Springer-Verlag (2008), ISBN 978-0-387-78900-2 , Theorem 2.17 and Theorem 2.19

[ME-01-1] Marcel Erné: Introduction to the theory of order. 1982, p. 230

[GG-01-2] George Graetzer: General Lattice Theory. 1978, p. 122

[ME-02-3] Erné, op.cit., P. 229

[4] This can be proven with the help of Zorn's lemma .

[5] Steven Roman: Lattices and Ordered Sets , Springer-Verlag (2008), ISBN 978-0-387-78900-2 , Theorem 2.17 and Theorem 2.19