On the generation of extensions of a partially ordered set

Abstract

A partially ordered set (or simply order) P is a set of elements E, together with a set R of relations of E, satisfying reflexivity, anti-symmetry and transitivity. The set E is called the ground set of P, while R is the relation set of it. There are many special orders. For example, when any two elements of E are related, the order is a chain. Similarly, we can define tree orders, forest orders and many others. An extension P’ of P is an order P’ having the same ground set as P, and such that its relation set contains R. When P’ is a chain then P’ is a linear extension of P. Similarly, when P’ is a forest then it is a forest extension of P. We consider the algorithmic problem of generating all extensions of a given order and also extensions of a special kind. The subject will be introduced by a general discussion on partially ordered sets.