Vol. 74, No. 2, 1978

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A solution for scattered order types of a problem of Hagendorf

Jean Ann Larson

Vol. 74 (1978), No. 2, 373–379

J. Hagendorf asked if every order type ϕ having the following two properties of additively indecomposable ordinals was the order type of an ordinal. Call ϕ Hagendorf if (i) it is strictly indecomposable to the right, i.e., if ϕ = ψ + 𝜃, then ϕ can be embedded in 𝜃 but not in ψ, and (ii) every strictly smaller type can be embedded in an initial segment of ϕ, i.e., if χ can be embedded in ϕ but not vice versa, then ϕ = ψ + 𝜃 where 𝜃0 and χ can be embedded in ψ. Recall that scattered order types are those which do not embed the order type of the rationals.

The paper provides a partial answer to Hagendorf’s question: Every scattered Hagendorf type is the order type of an indecomposable ordinal.

Other subclasses of order types for which this question seems particularly interesting are sub-types of the order type of the real numbers, and the class of countable unions of scattered types.

Mathematical Subject Classification 2000
Primary: 06A05
Received: 8 July 1977
Revised: 23 September 1977
Published: 1 February 1978
Jean Ann Larson