We show that the family of assignment matrices which give rise to the same nucleolus form a compact join-semilattice with one maximal element, which is always a valuation. -see p.43, Topkis, 1998-. We give an explicit form of this valuation matrix. The above family is in general not a convex set, but path-connected, and we construct minimal elements of this family. We also analyze the conditions to ensure that a given vector is the nucleolus of some assignment game.