In the multi criteria decision making context, let $\mathcal{A}$ be a set of alternatives or choices. Each alternative $\alpha\in \mathcal{A}$ is a vector of $k$ criteria $\alpha=(v_1,v_2,\dots,v_k)$. Let $\geq_1,\dots,\geq_k$ be a set of $k$ partial orders over $\mathcal{A}$. For any two alternative $\alpha,\beta\in \mathcal{A}$, $\alpha$ dominates $\beta$ $(\alpha\geq \beta)$ if and only if $\alpha\geq_i \beta$ for all $i\in\{1,2,\dots,k\}$.
An alternative $\alpha$ is said to be Pareto optimal if there is no alternative dominates it. That is, $\beta \not\geq \alpha$ for all $\beta\in \mathcal{A}$. The Pareto set $S_{pareto}$ is the set of all Pareto optimal alternatives.
I am looking for different notions with which one can have a subset of $S_{pareto}$. Are there other notions in the literature define a set $S$ which is guaranteed to be a subset of $S_{pareto}$?
