Acta Mathematiea Academiae Scientiarum Hungaricae Tomus 19 (3--4), (1968), pp. 311--327. ON THE COMPARISON OF ORDER TYPES By B. ROTMAN (Bristol) (Presented by P. ERD6S) Introduction Our purpose here is to study the comparison of order types (isomorphism types of ordered sets) with respect to the following relation of embeddability: if A, B are ordered sets with types a, fl respectively then a=<fl means that A can be embedded in B by an order preserving isomorphism. If we identify ~, fl when ~ <_- fl -<_ then any set of order types is partially-ordered by embeddability. In the special case when the order types are those of well-ordered sets, i.e. ordinals, the embedda- bility relation is the usual ordering relation between ordinals and any set of ordinals is well-ordered by it. In the general case of arbitrary order types, however, there exist incomparable types, i.e. types e, fl such that a~fl~a and it is the existence of such types which complicates the investigation of the embeddability relation. The above definition of embeddability, which is a natural one, seems to have been first suggested for order types by FRA~SSt~ in [4]. In [4], FRA[SSt~ announces certain results and poses several questions concerning the embeddability relation for denumerable order types. In view of the fact that the methods used to investigate the embeddability relation in the case of denumerable types differ markedly from those used in the case of non-denumerable types.we have restricted ourselves here to one case only, namely the case of non-denumerable types; in a later publication (see footnote 13 below) we shall present some results on denumerable order types. The embeddability relation for non-denumerable types has so far been investi- gated only in the case of types of the power of the continuum; first by SIERPINSKI in [14] and then by GINSBURGin [7], [8], [9]. In fact both authors confine themselves to types of the power of the continuum which can be embedded in the continuum. Our main concern here is to show that if m is an arbitrary infinite cardinal and if the Generalised Continuum Hypothesis is assumed whenever necessary then almost all the results of Sierpinski and Ginsburg hold for certain types of power m + which can be described roughly as those which have subtypes dense in them of smaller power. Preliminaries The cardinal number of a set X will be denoted by IXI and its order-type if it is ordered by tp(X). Inclusion between sets will be denoted by c in the strict sense and by ~ in the wider sense. The letters i, j, k will denote ordinals and m, n, r cardinals; no distinction will be made between finite ordinals and cardinals. Lower case Greek letters will denote arbitrary order-types where we reserve 2 for the type of the continuum and t/ for the type of the rationals. A similarity map- ping is an order preserving (1, 1) function. If (A, <) is an ordered set of type then the reserve type ~* denotes the type of the ordered set (A, >-). IfXis an ordered Acta Matbematica Academiae Scientiarunz Hungaricae z9, i968