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

In this contribution, we aim to introduce and study L-fuzzy partition spaces and L-fuzzy closure system spaces in a categorical framework. Further, we present the concepts of coalgebras and dialgebras corresponding to a direct upper F -transform under certain conditions and show the functorial relationship between the category of spaces with L-fuzzy partition and the category of coalgebras (dialgebras). Moreover, we show that the categories of coalgebras and dialgebras are isomorphic and introduce a pair of adjoint functors between the coalgebras and dialgebras.