Aggregation of First-Type L-Fuzzy Rough Approximation Operator

First-type L-fuzzy rough approximation operators have so far been studied relative to a single L-fuzzy relation on a universe of discourse, an assumption at odds with the way multi-source, multi-sensor, and multi-granular information systems are actually modelled. This paper analyses their aggregation over an arbitrary indexed family of relations, combining the individual upper approximations by pointwise infimum and the individual lower approximations by pointwise supremum. We show that this meet/join rule is the unique aggregation mechanism compatible with a natural conservative axiom system, and we exhibit, on an explicit finite completely distributive lattice, a family of relations whose aggregated operators cannot be reproduced by any single relation. This is an existence result: a second example on the three-element chain shows that the phenomenon is not peculiar to one lattice, while on the two-element Boolean lattice every pair of relations on a two-point universe turns out to be representable, so our examples do not establish non-representability on every completely distributive lattice. This non-representability turns out to be an operator-level phenomenon: the associated fixed-point systems can coincide with those of a single, universal relation even when the operators themselves do not, so that representability at the level of operators and at the level of fixed points are logically independent properties. We further establish the monotonicity bounds satisfied by the aggregated operators under arbitrary unions and intersections of inputs, and show by a finite counterexample that complete distributivity of the underlying lattice does not strengthen these bounds into equalities. Building on these operator-level results, we identify explicit conditions under which the fixed-point system of an aggregated operator forms an Alexandrov fuzzy topology, exhibit a family whose fixed-point system is strictly richer than that of a specific single relation, and delimit the separation and connectedness properties that such fixed-point systems may or may not inherit from their constituent relations. Finally, we use the aggregated operators for three-way decisions and compare them with output averaging and relation-level fusion through a worked example, a sensitivity analysis and a simulation study. Among all rules that aggregate the outputs of the sources internally, the conservative rule defers the fewest objects, for the minimum and maximum it never commits against the object’s own evidence when the thresholds lie on either side of 1/2, and its price is that it inherits the errors of every source.