1. Examples

#1. In a diagram of the free distributive lattice    FDL$(3)$ (Figure ), if the generators are $g_1,g_2,g_3$ you can see that

$(g_1 \vee g_2) \wedge (g_1 \vee g_3) \wedge (g_2 \vee g_3) = (g_1 \wedge g_2) \vee (g_1 \wedge g_3) \vee (g_2 \wedge g_3)$.

Once it is known that this lattice is indeed a free distributive lattice on three generators, then it follows that this law holds in all distributive lattices:

$(x _ 1 \vee x _ {2}) \wedge (x _ 1 \vee x _ {3}) \wedge (x _ 2 \vee x _ {3}) = (x _ 1 \wedge x _ {2}) \vee (x _ 1 \wedge x _ {3}) \vee (x _ 2 \wedge x _ {3})$

Figure: FDL(3)

text/Bdir/fdl3.eps

#2. The free Boolean algebra FBA$(3)$, corresponding to a Venn diagram with three circles. It has 8 atoms and 256 elements.

#3. The free modular lattice FML$(3)$ shown in Figure . It has 28 elements.

Figure: FML(3)

text/Bdir/fml3.eps

#4. The free lattice FL$(3)$ shown in Figure . It is infinite. Dashed lines represent infinitely many elements not shown.

Figure: FL(3)

#5. The free abelian group on $n$ generators is Z$^ n$.

#6. The free group FG$(2)$ consists of all finite expressions such as $g ^ 2 h ^ {-3} g h ^ 2$, with appropriate equalities.

#7. Every vector space is free, with generators being any basis.

#8. For a given type $\tau$, the term algebra $T _ {\tau}(n)$ is the set of all $n$-ary terms of type $\tau$, with operations being formal compositions. The generators are the variable symbols $x _ 1,\dots, x _ n$.