this paper extends the notion of ditopology to the case where openness and closedness are given in terms of {em a priori} unrelated drading functions. the resulting notion of graded ditopology is considered both in the setting of lattices and in that textures, the relation between the two approaches being discussed in detail. interrelations between graded ditopologies and ditopologies on textur...

the main purpose of this paper is to introduce a concept of$l$-fuzzifying topological groups (here $l$ is a completelydistributive lattice) and discuss some of their basic properties andthe structures. we prove that its corresponding $l$-fuzzifyingneighborhood structure is translation invariant. a characterizationof such topological groups in terms of the corresponding$l$-fuzzifying neighborhoo...

employing the notions of the strong $l$-topology introduced by zhangand the $l$-frame introduced by yao  and the concept of $l$-enrichedtopological system defined in the present paper, we constructadjunctions among the categories {bf st$l$-top} of strong$l$-topological spaces, {bf s$l$-loc} of strict $l$-locales and{bf $l$-entopsys} of $l$-enriched topological systems. all of theseconcepts are ...

The paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. Motivated by an open question of S. E. Rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of L. A. Zadeh. On the next step, w...