Higher Preferential Structures

Abstract

Recall from Chapter 1 that the basic property of “normal” preferential structures is that minimization is upward absolute. If $\mathit{x, y} \in \mathit{X,\, x} \prec \mathit{y},\,\, \mathrm{i.e.,} \,\, \mathit{y}$ is a non-minimal element in X, and $\mathit{X} \subseteq \mathit{Y}$ then y will be a non-minimal element in Y, too - as $\mathit{x} \in \mathit{Y}$ This results in the fundamental “algebraic” property that for $\mathit{X} \subseteq \mathit{Y} \,\,\mu (\mathit{Y}) \cap \mathit{X} \subseteq \mu (\mathit{X}), \mathrm{where} \,\,\mu (\mathit{X})$ is the set of minimal elements of X.