Naturally Ordered Orthodox Dubreil-Jacotin Semigroups

The structure of orthodox semigroups with a normal band of idempotents has been described by Yamada. Since, for a naturally ordered strong Dubreil-Jacotin orthodox semigroup, it can be shown that the band of idempotents is normal, it is of interest to investigate the structure of these ordered orthodox semigroups via the Yamada decomposition. What is hard in such matters, and no exception here, is how to marry together the order theoretic structure with the algebraic structure theory. Basically, the problem is how to define orders on the building bricks of the structure theory in such a way that the algebraic isomorphisms become order isomorphisms. This we are able to do for a variety of different types, obtaining structure theorems which concern not only cartesian orders but also several new types of lexicographic orders. Examples are given to illustrate the hierarchy of orders and the corresponding algebraic conditions required for the order isomorphisms.