A *pearl* is an ordered pair $\ \mathbf P:=(G\,\ S),\ $ where $\ G\ $ is a group, and $\ S\ $ is a non-empty subset of G which does not contain the neutral element of $\ G\ $ (i.e. not 1 in the multiplicative notation, nor 0 in the additive notation).

If $\ \mathbf Q:=(H\,\ T)\ $ is another pearl then a morphism
$\ f:\mathbf P\rightarrow \mathbf Q\ $ is defined as a group homomorphism
$\ f:G\rightarrow H\ $ such that $\ f(S) \subseteq T$.

This defines the (general) category of pearls. Pearls for which their group is Abelian, form a full subcategory (with all their morphisms between them).

Let's define a free pearl $\ \mathbf P\ $ (as above) as one for which a subset $\ F\subseteq G\ $ is a set of the free generators of the group $\ G,\ $ and $\ S\subseteq F.\ $ Obviously, every free pearl is projective (in the category of pearls).

The definition of an abelian-free pearl (and the issue of projectivity) is similar as for the category of abelian pearls.

**QUESTION**   Can you provide examples of projective pearls which are not free? The same for the case of the category of abelian pearls.

>**REMARK** This requires paying close attention to pearl epimorphisms.

                **M O T I V A T I O N**

1. Categories of enhanced groups are among the most important. After groups, pearls are among the simplest among them. Thus it provides a view onto the next categories which already appear;

2. There are problems on the general categories (or a general class of them) when we would like to obtain a positive result or else a counter-example. Then pearls may provide a relatively simple testing ground and they may project onto other categories.

3. Truly, the notion of pearls appeals to me, I find them attractive.