$r$-connectivity of $L$-subsets in convex $(L,M)$-fuzzy hull spaces

In general topology, connectedness is a fundamental concept. Convex structures, analogous to topological structures, enable the generalization of numerous properties from topological spaces. In this paper, we first define $r$-connectivity for $L$-subsets in convex $(L,M)$-fuzzy hull spaces based on convex $(L,M)$-fuzzy hull operators. We then establish several equivalent characterizations of $r$-connected $L$-subsets and explore their key properties. In particular, by leveraging these operators, we introduce the finite product property of $r$-connecivity for product spaces within the framework of convex $(L,M)$-fuzzy hull spaces.