---- > [!definition] Definition. ([[semigroup]]) > A **semigroup** is a pair $(G, \cdot)$ consisting of a set $G$ and a [[binary operation]] $\cdot: G \times G \to G$ that is [[associative]]. > \ > Semigroups may be considered generalizations of [[group|groups]] that remove the requirements of identity and inverses. > [!basicexample] > $\mathbb{N}$ forms a [[semigroup]] under addition. ^basic-example ---- #### ---- #### 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 > ```