Ordered categories and ordered semigroups

We use ordered categories to study semidirect decompositions of finite ordered semigroups. We obtain ordered analogues of the derived category theorem and of the delay theorem. Next we prove that the ordered analogues of semilattices and of J -trivial monoids constitute local varieties, and we derive some decomposition theorems from these results. In [10], Pin and Weil initiated a study of the semidirect decomposition of finite ordered semigroups. We refer the reader to the introduction of that article for details on the motivations for such a study, especially in connection with formal language theory. In another paper [11], Pin and Weil give some applications of semidirect decomposition results to language theory, some of which depend on results from the present article. The foundations for the study of the semidirect product of ordered semigroups, are given in [10], as are some interesting decomposition results for classes of finite ordered semigroups, notably regarding naturally ordered inverse monoids, ordered monoids in which the unit is the maximum element, and ordered monoids in which the unit is maximum among the idempotents. In order to go further towards decomposing varieties of ordered semigroups and monoids, we need to extend our scope and to consider finite ordered semigroupoids and categories, just like in the decomposition theory of unordered semigroups and monoids. Semigroupoids are defined like LIAFA, Université Paris VII and CNRS, Case 7014, 2 Place Jussieu, 75251 Paris Cedex 05, France LaBRI, Université Bordeaux I and CNRS, 351 cours de la Liberation, 33405 Talence Cedex, France Work supported by INTAS project 1224.