# Hausdorff ’s theorem for posets that satisfy the finite antichain property

Fundamenta Mathematicae (1999)

• Volume: 159, Issue: 1, page 51-69
• ISSN: 0016-2736

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## Abstract

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Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC).  Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations: (1) inversion, (2) lexicographic sum, and (3) augmentation (where $⟨P,⪯⟩$ augments ⟨P, ≤⟩ iff $x⪯y$ whenever x ≤ y). We show that this closure consists of all scattered posets satisfying the

## How to cite

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Abraham, Uri, and Bonnet, Robert. "Hausdorff ’s theorem for posets that satisfy the finite antichain property." Fundamenta Mathematicae 159.1 (1999): 51-69. <http://eudml.org/doc/212319>.

@article{Abraham1999,
abstract = {Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC).  Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations: (1) inversion, (2) lexicographic sum, and (3) augmentation (where $⟨P, \preceq ⟩$ augments ⟨P, ≤⟩ iff $x \preceq y$ whenever x ≤ y). We show that this closure consists of all scattered posets satisfying the},
author = {Abraham, Uri, Bonnet, Robert},
journal = {Fundamenta Mathematicae},
keywords = {ordinals; partial orderings; scattered partial orderings; Hausdorff's theorem; scattered sets; finite antichain property; antichain rank; well-founded posets; product operation for ordinals},
language = {eng},
number = {1},
pages = {51-69},
title = {Hausdorff ’s theorem for posets that satisfy the finite antichain property},
url = {http://eudml.org/doc/212319},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Abraham, Uri
AU - Bonnet, Robert
TI - Hausdorff ’s theorem for posets that satisfy the finite antichain property
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 1
SP - 51
EP - 69
AB - Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC).  Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations: (1) inversion, (2) lexicographic sum, and (3) augmentation (where $⟨P, \preceq ⟩$ augments ⟨P, ≤⟩ iff $x \preceq y$ whenever x ≤ y). We show that this closure consists of all scattered posets satisfying the
LA - eng
KW - ordinals; partial orderings; scattered partial orderings; Hausdorff's theorem; scattered sets; finite antichain property; antichain rank; well-founded posets; product operation for ordinals
UR - http://eudml.org/doc/212319
ER -

## References

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1. [1] U. Abraham, A note on Dilworth's theorem in the infinite case, Order 4 (1987), 107-125. Zbl0629.06002
2. [2] R. Bonnet et M. Pouzet, Extension et stratification d'ensembles dispersés, C. R. Acad. Sci. Paris Sér. A 168 (1969), 1512-1515. Zbl0188.04203
3. [3] P. W. Carruth, Arithmetic of ordinals with applications to the theory of ordered abelian groups, Bull. Amer. Math. Soc. 48 (1942), 262-271. Zbl0061.09308
4. [4] R. Fraïssé, Theory of Relations, Stud. Logic Found. Math. 118, North-Holland, 1986.
5. [5] F. Hausdorff, Grundzüge einer Theorie der geordneten Mengen, Math. Ann. 65 (1908), 435-505. Zbl39.0099.01
6. [6] G. Hessenberg, Grundbegriffe der Mengenlehre, Abh. Friesschen Schule neue Folge 4 (1906).
7. [7] A. Lévy, Basic Set Theory, Springer, 1979.
8. [8] W. Sierpiński, Cardinal and Ordinal Numbers, Monograf. Mat. 34, PWN, Warszawa, 1958.

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