Closure structures parameterized by systems of isotone Galois connections

We study properties of classes of closure operators and closure systems parameterized by systems of isotone Galois connections. The parameterizations express stronger requirements on idempotency and monotony conditions of closure operators. The present approach extends previous approaches to fuzzy closure operators which appeared in analysis of object-attribute data with graded attributes and reasoning with if-then rules in graded setting and is also related to analogous results developed in linear temporal logic. In the paper, we present foundations of the operators and include examples of general problems in data analysis where such operators appear.