On stratified L-convergence spaces: Pretopological axioms and diagonal axioms

This paper focuses on pretopological and diagonal axioms for stratified L-convergence spaces recently defined by the authors in 2011. This notion is slightly stronger than Jäger's stratified L-generalized convergence spaces. In particular, one pretopological axiom is stated without resorting to the notion of neighborhood L-filters; some generalizations of Kowalsky's and Fischer's diagonal conditions are presented. The relationship between these axioms are investigated.