Holes in ordered sets (Q1084421)

Using the technique of ''holes'' the authors study the retracts of an ordered set. As an application a close connection between the class of absolute retracts and the class of dismantlable ordered sets is established. The main results are the following: I. A connected ordered set is a retract of a direct product of fences if and only if each hole can be separated on a fence; II. The class of all absolute retracts is a variety; III. Every finite connected absolute retract is dismantlable by irreducibles; IV. Every finite ordered set which is dismantlable by doubly irreducibles is itself an absolute retract.