On the embedding of convex spaces in stratified L-convex spaces

Consider L being a continuous lattice, two functors from the category of convex spaces (denoted by CS ) to the category of stratified L -convex spaces (denoted by SL - CS ) are defined. The first functor enables us to prove that the category CS can be embedded in the category SL - CS as a reflective subcategory. The second functor enables us to prove that the category CS can be embedded in the category SL - CS as a coreflective subcategory when L satisfying a multiplicative condition. By comparing the two functors and the well known Lowen functor (between topological spaces and stratified L -topological spaces), we exhibit the difference between (stratified L -)topological spaces and (stratified L -)convex spaces.