Theory of operators and Unitary operators on the power set and the way below relation on transitive binary relational sets

In this work, S. Abramsky and A. Jung introduced a method to construct a canonical partially orderd set from a pre-order set. The main goal of this monograph is to propose a new representation theory involving below relation, lower ( resp. upper) closure, upper (resp. a lower) cone, convex hull, cofinal set, directed set, domain set and Scott-open (resp Scott-closed) set. We also study the\ family\ of all isolated points above (resp. below), isolated point \ in transitive relational sets. Order preserving monotone and idempotent function between two posets are introduced and discussed, where the latter become representable in all cases, and still rich enough to allow geometric, topological and combinatorial applications. Throughout the text, we shall give evidence of the geometric potential of these new ideas. These concepts extend in many aspects of the known geometric theory of poset, but they also raise new perspectives in the topological world. In particular, we believe that our results and techniques
may be of interest in connection with several of the famous conjectures and constructions for algabra and topology. The readers consider them as a source of new concepts, techniques and problems for algabric theory.