  We investigate relations on elements in C*-algebras, including *-polynomial
relations, order relations and all relations that correspond to universal
C*-algebras. We call these C*-relations and define them axiomatically. Within
these are the compact C*-relations, which are those that determine universal
C*-algebras, and we introduce the more flexible concept of a closed
C*-relation.
  In the case of a finite set of generators, we show that closed C*-relations
correspond to the zero-sets of elements in a free sigma -C*-algebra. This
provides a solid link between two of the previous theories on relations in
C*-algebras.
  Applications to lifting problems are briefly considered in the last section.
