A partial order that is not rank/trace complete

Abstract

Let P be a preorder relation on a finite set G. The algebra CG×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∈CG×G[G] is said to satisfy the P-rank/trace conditions ifrankM[J|J]⩽traceM[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 CG×G[G]. In this note, we provide an example of a partial order that is not rank/trace complete.