A structure theory for ordered sets

The theory of ordered sets lies at the confluence of several branches of mathematics including set theory, lattice theory, combinatorial theory, and even aspects of modern operations research. While ordered sets are often peripheral to the mainstream of any of these theories there arise, from time to time, problems which are order-theoretic in substance. The aim of this work is to fashion a classification scheme for ordered sets which aimed at providing a unified vantage point for some of the problems encountered with ordered sets. This classification scheme is based on a structure theory much akin to the familiar subdirect representation theory so useful in general algebra. The novelry of the structure theory lies in the importance that we attach to, and the widespread use that we make of, the concept of retract. At present, some vindication for our classification scheme can be found by examining its effectiveness for totally ordered sets (e.g., well-ordered sets, the rationals, reals, etc.) as well as for those finite ordered sets that arise commonly in combinatorial investigations (e.g., crowns and fences).

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