General L-fuzzy aggregation functions based on complete residuated lattices

As a vital tool in data analysis, aggregation functions have been widely studied in many papers. In particular, one of the recent research topics for aggregation functions is the study of the various extension forms of those useful functions. This paper continues to research this topic from the theoretical point of view. First, we introduce the notions of L-fuzzy aggregation functions and general L-fuzzy aggregation functions based on complete residuated lattices. Then we present the upper and lower general L-fuzzy aggregation approximation functions of the general L-fuzzy aggregation functions, which are the pointwise extension of an L-fuzzy aggregation function. Moreover, we consider some vital properties of those aggregation approximation functions and investigate the relationship between those aggregation approximation functions and the corresponding L-fuzzy relations. Finally, we show that the approach of axiomatizations of the upper and lower general L-fuzzy aggregation approximation functions ensures the existence of corresponding L-fuzzy relations which generate the functions.