0. The concept

Definition. Let $V$ be a variety. The algebra $F$ is free in $V$ on $g _ 1,\dots, g _ n$ if

(i) $F \in V$,

(ii) $F$ is generated by $g _ 1,\dots, g _ n$, and

(iii) the only term relations holding between $g _ 1,\dots, g _ n$ are those that hold for all $n$-tuples in all algebras in $V$, i.e., are the laws holding in $V$.

(In examples generators may also be labeled $g,h,k$ or $a,b,c$, etc.)