The problem of invariance of keys including simple keys is investigated as applied to operations of table algebras that are modern analogues of classical relational Codd algebras. It is shown that the keys being considered are invariant with respect to the operations of intersection, difference, selection, join, and division, but invariance does not hold for simple keys. The keys including simple keys are shown to be invariant with respect to the renaming operation. Necessary and sufficient conditions under which such keys including simple keys are invariant with respect to the operations of projection and active supplement are established. The results of this work are of theoretical and practical interest and can be used to choose optimal keys in designing relational databases.