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

Gähler's neighborhood condition for lattice-valued convergence spaces

“…It appeared first for the case that L is a frame in [3] and a space satisfying this axiom was called a stratified L-ordered convergence space. In the case that L is a residuated lattice, this axiom was used in [23,24]:…”

“…We can characterize the axiom (LK) in the category SL-PCS as follows. [24]). Let (X, lim) ∈ |SL.PC S|.…”

“…Later, Yao [33] generalized lattice-valued convergence spaces to the lattice context of complete residuated lattices. A similar lattice background is also used in [23,24], however, the definitions of a lattice-valued filter in these two papers do not coincide. Looking at the definition of a stratified lattice-valued filter in [13], it becomes clear that both papers actually use different enriched cl-premonoids.…”

“…In [16,17] we studied generalizations of Kowalsky's and of Fischer's diagonal conditions in the category CONV of convergence spaces. In [24] similar diagonal conditions, in a stronger axiom scheme, were considered. This paper studies a different condition.…”

“…The first one is suitable to characterize lattice-valued topological spaces in a subcategory of SL-GCS. Objects in this subcategory were called ordered stratified L-convergence spaces in [3] (for L a complete Heyting algebra) and they were also considered, in a more general lattice-context, in [23,24]. The other generalization of Gähler's neighborhood condition is a 'levelwise condition'.…”