Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set,  determine the covering relation \prec of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C' where C and C' are two such closure systems satisfying C \prec C'.  also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements.
['Nathalie Caspard', 'Bernard Monjardet']
I am writing a research paper on a idea Some lattices of closure systems on a finite set
[{"from": "human", "value": "I am writing a research paper on a idea Some lattices of closure systems on a finite set"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set,  determine the covering relation \\prec of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C' where C and C' are two such closure systems satisfying C \\prec C'.  also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements."}]