Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...

The notation and terminology used in this paper have been introduced in the following articles: [3], [19], [9], [10], [16], [20], [4], [5], [22], [23], [21], [1], [6], [17], [18], [2], [25], [26], [24], [15], [12], [13], [8], [14], and [7]. Let A be a non empty set, x be an element, and a be an element of A. Let us observe that (A 7−→ x)(a) reduces to x. Let A, B be non empty topological spaces...

Fuzzy relations are mappings from pairs of elements into the interval [0, 1]. As a replacement for the complement operation one can use the mapping that sends x to 1 − x. Together with the concepts of t-norm and t-conorm a weak form of Boolean algebra can be defined. However, to our knowledge so far no notion of domain or codomain has been investigated for fuzzy relations. These might, however,...