---- > [!definition] Definition. ([[word on a set]]) > Let $A$ be a set and $A'$ an [[bijection|isomorphic]] copy of $A$; call $a ^{-1} \in A'$ the element in $A'$ corresponding to $a \in A$. > > A **word of length $n$ on the set $A$** is an ordered list $(a_{1},a_{2},\dots,a_{n})$ > which we denote by the juxtaposition $w=a_{1}a_{2}\cdots a_{n}$ > where each 'letter' $a_{i}$ is either an element $a \in A$ or an element $a ^{-1} \in A'$. > \ > The set of words on $A$ is denoted $W(A)$. Included in $W(A)$ is the **empty word** $()$ consisting of 'no letters'. ---- #### - clarification on [[finite direct products and coproducts align in the category of abelian groups]] - and insight on the related exercise - 5.6 ---- #### 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 > ```