---- > [!definition] Definition. ([[upper set]]) > Let $(X, \leq)$ be a [[poset|preordered set]]. An **upper set** in $X$ is a subset $U \subset X$ that is "closed under going up", in the sense that for all $u \in U$ and all $x \in X$, $u \leq x \implies x \in U.$ > The [[opposite category|dual]] notion is that of a **lower set** in $X$, a subset $L \subset X$ that is "closed under going down", in the sense that for all $\ell \in L$ and all $x \in X$, $x \leq \ell \implies x \in L.$ Lower sets are sometimes called **order ideals**. ^definition ---- #### Definition from Wikipedia. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```