On constructive models of theories with linear Rudin-Keisler ordering

It is known that the class of Ehrenfeucht theories admits a syntactical characterization and that a finite (Rudin-Keisler) pre-ordering and a function mapping this pre-ordering to naturals play the role of parameters in this characterization. In the article, we construct for any finite linear ordering L, a hereditary decidable Ehrenfeucht theory T possessing L as its Rudin-Keisler pre-ordering. Also, we discuss decidable and computable models of such theories.