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In this contribution, we aim to introduce and study L-fuzzy partition spacesand L-fuzzy closure system spaces in a categorical framework. Further, wepresent the concepts of coalgebras and dialgebras corresponding to a directupper F -transform under certain conditions and show the functorialrelationship between the category of spaces with L-fuzzy partition and thecategory of coalgebras (dialgebras). Moreover, we show that the categories ofcoalgebras and dialgebras are isomorphic and introduce a pair of adjointfunctors between the coalgebras and dialgebras.

On categories of spaces with L-fuzzy partitions, L-fuzzy closure system spaces and coalgebras (dialgebras)