Universal semiorders

A Z-product is a modified lexicographic product of three total preorders such that the middle factor is the chain of integers equipped with a shift operator. A Z-line is a Z-product having two linear orders as its extreme factors. We show that an arbitrary semiorder embeds into a Z-product having the transitive closure as its first factor, and a sliced trace as its last factor. Sliced traces are modified forms of traces induced by suitable integer-valued maps, and their definition is reminiscent of constructions related to the Scott–Suppes representation of a semiorder. Further, we show that Z-lines are universal semiorders, in the sense that they are semiorders, and each semiorder embeds into a Z-line. As a corollary of this description, we derive the well known fact that the dimension of a strict semiorder is at most three.