Embedding Theorems for Multiplicative Systems and Projective Geometries

Introduction. It has been shown in a recent paper (see [5]) that any countable group can be embedded in a group generated by two elements. We show here that any countable loop (quasigroup, groupoid) can be embedded in a loop (quasigroup, groupoid) generated byone element, any countable semigroup can be embedded in a semigroup generated by two elements, and any countable projective plane can be embedded in a projective plane generated by four points. No such embedding theorem exists for some systems, such as abelian groups, and some light is thrown on the general problem by the following theorem. Let 21 be a class of algebras, (Ei) the property that there is a number m such that any countable 2l-algebra can be embedded in an 2l-algebra generated by m elements, (E2) the property that there is a number n such that the free 2I-algebra on n generators contains a free subalgebra on a countable number of generators. Then if 21 has property (Ei), it has also property (E2) and n^m.