It is easy to see that every infinitely countable linearly ordered set embeds into Q (rational numbers) with its linear order.("Linear order"="total order". All embeddings are assumed to preserve linear orders). In fact, we expect that Q is a "minimal" linearly ordered set with this property.
What about sets of higher cardinalities? Specifically, is it true that for any cardinality X, there is a linearly ordered set S, which is "minimal" with respect to the property that every linearly ordered set of cardinality X embeds into S?
What is S for X=continuum?