I am looking into the book "Inequalities: Theory of Majorization and Its Applications, second ed. " of Marshall and Olkin.
In chapter 14 (Ordering Extending Majorization, section E) the definition of orderings derived from function sets is given as:
Let $\Phi$ a set of real-valued functions $\phi$ defined on a subset $A$ of $R^d$. For $x,y\in A$, we write $x\prec_\Phi y$ to mean that: $$ \phi(x)<\phi(y),\;\;\forall\;\phi\in\Phi. $$ The section goes on giving some few examples.
I am interested if, in this more general context, it is possible to derive an equivalent condition then the one given by Hardy, Littlewood and Polya for the majorization order in terms of doubly stochastic matrices: $$x\prec y\;\;\;iff\;\;\; x=Dy,\;\;\exists\; D\in DS(d)$$ i.e.: $$x\prec_\Phi y\;\;\;iff\;\;\; x=Ty$$ exists $T$ with $T$ a matrix in some set of matrices to be characterized from the set $\Phi$.
More in general I am interested in other examples in literature for such definition of a partial order. I have not been able to find much.
Thanks a lot!
Fabio
