Maximal chains and antichains in strongly noetherian semiorders

In the field of abstract measurement theory, semiorders have assumed an especially important role as a natural setting for expressing the comparative relations among the magnitudes of measurands. If we assume in addition that our measuring scale is bounded below, which happens in many important cases, we get an interesting class of semiorders, the strongly noetherian ones. In this work, we study some combinatorial and order-theoretic properties of strongly noetherian semiorders, with a particular regard to the structure of maximal chains and antichains (called lines and cuts in the present paper). We see that the cuts determine a sort of dynamics in the semiorder, which may be described as a linear order on the cuts themselves; from this linear order one can extract the main properties of the original semiorder, whose elements are represented as closed intervals of cuts. Moreover, "continuity" properties such as K-density and coherence have a natural description in term of simple cut properties; as a corollary, we obtain that K-density is equivalent to N-freeness in strongly noetherian semiorders.