Fischer's Diagonal Condition for Lattice-Valued Convergence Spaces

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

“…It should be stressed that the axiom (Lf) ensures that a stratified L-generalized convergence space is topological. In contrast, as observed in [14,24], the axiom (Lfw) does not work in this regard (even in our stronger axiom scheme, see Example 4.7).…”

“…L X was given for L-filters (where the "intersection axiom" is different) in [9,Section 2.4.2]. k L appeared in [14], k L (F) = G(F (.) ), when * = ∧.…”

“…In particular, the axiom (Lq) enables us to formulate pretopological axiom without using the notion of neighborhood L-filter, which plays a dominative role in Refs. [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 :…”

“…For a stratified L-generalized convergence space (X, lim), Jäger [14] proposed the condition (Lfw) Let J be any set, : J − → X and : J −→ F s L (X ) be mappings. Then for all F ∈ F s L (J ) and x ∈ X ,…”

“…In the case that L being a complete Heyting algebra, Jäger [13,14] introduced a generalized Kowalsky's diagonal axiom (Lk) and a generalized Fischer's diagonal axiom (Lfw) for stratified L-generalized convergence spaces. The two axioms can be easily generalized to the setting that L being a complete residuated lattice.…”