On adjunctions between Lim, SL-Top, and SL-Lim

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

“…[11][12][13][14]. [20] (or, a stratified L-ordered convergence structure in the terminology of [2]) on a set X is a function lim :…”

“…The following example shows that this is not true for stratified L-generalized convergence spaces in general. [2], Jäger [13], Li and Jin [20]). Let L be the linearly ordered frame ({0, , 1}, ∧, 1) with 0 < < 1.…”

“…In [2,20], it is proved that (X, lim) is a stratified L-generalized convergence space but it does not satisfy (LC2), so, it does not fulfill (Lp) since (Lp) ⇒ (LC2). However, it is easily seen that ∀z ∈ X, lim U z lim (z) = lim[z](z) = 1.…”

“…In [18,20] the interrelationship between limit spaces, stratified L-limit space and stratified L-topological spaces is investigated. In particular, it is observed that a stratified L-convergence structure is naturally a stratified L-generalized convergence structure but the converse is not true.…”

Gähler's neighborhood condition for lattice-valued convergence spaces

“…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.…”