3. Existence of free algebras in $V =$   Var$(A)$

Let the free algebra on $n$ generators in Var$(A)$ be denoted $F _ A(n)$.

Theorem (Birkhoff) $F _ A(n)$ can be constructed as follows:

Let $\Delta$ be the set of all functions $\delta: \{1,\dots n\} \rightarrow A$, and let $P = A ^ {\Delta}$.

For $i =1,\dots, n$ let $g _ i \in P$ be the element whose $\delta$-th coordinate is $\delta(i)$.

Let $F$ be the subalgebra of $P$ generated by $g _ 1,\dots, g _ n$.

Then $F = F _ A (n)$.

Example. To generate $F _ {\mbox{\bf 2}} (3)$ (= FDL$(3)$), where 2 is the 2-element lattice, proceed as shown in Figure .

Figure: Construction of FDL(3) as $F _ {\bf 2}(3)$

As another example, Figure shows the table obtain for $A =$   Z$_ 3$ under subtraction and for $n = 2$:

Figure: Construction of $F _ {\mbox{\bf Z} _ 3}(2)$ under subtraction

The rows form the free algebra $F _ A (2)$ inside $A ^ 9$. Of course, this example is really a disguised version of an additive group.