Let P be a preorder relation on a finite set G. The algebra C G × G [ G ] consists of all complex matrices (with rows and columns indexed by G) which have zeros at those positions ( i , j ) which are not in P . A subset J ⊂ G is called P -convex if the conditions a , c ∈ J , ( a , b ) ∈ P , ( b , c ) ∈ P imply b ∈ J . A matrix M ∈ C G × G [ G ] is said to satisfy the P -rank/trace conditions if rank M [ J | J ] ⩽ trace M [ J | J ] ∈ Z + holds for the restriction M [ J | J ] of M to any P -convex set J. A preorder P is called rank/trace complete if any matrix satisfying the P -rank/trace conditions is a sum of rank-one idempotents in C G × G [ G ] . In this note, we provide an example of a partial order that is not rank/trace complete.
