Algebras stratified for all linear orders

In this paper we describe several characterizations of basic finite-dimensional $k$-algebras $A$ stratified for all linear orders, and classify their graded algebras as tensor algebras satisfying some extra property. We also discuss whether for a given preorder $\preccurlyeq$, $\mathcal{F} (_{\preccurlyeq} \Delta)$, the category of $A$-modules with $_{\preccurlyeq} \Delta$-filtrations, is closed under cokernels of monomorphisms, and classify quasi-hereditary algebras satisfying this property.