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Paracompactness for Ordered Sums

Let us say that an order embedding of an uncountable regular cardinal in a linearly ordered set is continuous if it preserves the suprema (for all smaller limit ordinals). This makes the embedding a homeomorphism for the two order topologies; and if the image has no supremum, it is a closed subspace. Since uncountable regular cardinals fail to be paracompact, a linearly ordered set can be paracompact only if it admits no such embedding or anti-embedding. Conversely, Gillman and Henriksen have shown that this suffices (Trans. A.M.S. 77 (1954) pp. 352 ff).