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

• Autorzy: Uri Abraham , Robert Bonnet
• Języki publikacji: EN

Abstrakty

EN

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

Słowa kluczowe

EN

ordinals; partial orderings; scattered partial orderings; Hausdorff's theorem

Kategorie tematyczne

• 06B05: Structure theory
• 54E45: Compact (locally compact) metric spaces
• 06B30: Topological lattices, order topologies

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 159
• Numer: 1
• Strony: 51-69

Daty

wydano

1999

otrzymano

1997-01-16

poprawiono

1998-06-25

Twórcy

autor

Uri Abraham

• Department of Mathematics and Computer Science, Ben-Gurion University, Be'er Sheva, Israel

autor

Robert Bonnet

• Laboratoire de Mathématiques, Université de Savoie, 73011 Chambéry, France

Bibliografia

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