---- > [!definition] Definition. ([[ascending chain condition]]) > A [[poset]] $P$ is said to satisfy the **ascending chain condition** provided that every ascending [[poset|chain]] $p_{1}<p_{2}<p_{3}< \dots $ > stabilizes: there exists $i \in \mathbb{N}$ such that $p_{i}=p_{i+1}=p_{i+2}=\cdots.$ > > The **descending chain condition** may be defined analogously. ^definition ---- #### ---- #### 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 > ```