Conventional R-operations generate singularities in intersection points of boundaries of primitive domains. Some singularities are intrinsic and can not be removed. Others are induced by choosing inappropriate primitives, mathematical expressions and/or order of R-operations and they will be named ``parasitic singularities'' (PSg). In this paper, the R-functions without PSg will be referred to as ``Clean R-functions'' (CRF). Some modifications to the R-function method and some ideas and strategies for CRF generation are introduced in this paper. A new extension to R-operations is the R3-disjunction of three arguments. Unfortunately, these ideas are not general, but specific to a particular classes of problems. In conclusion, some questions about unexplored subtopics and suggestions for future research are given. The need to direct research towards generalisation is emphasized. This paper has an apparently relaxed style, but it has a serious intention to draw reader's attention on some only partially solved problems. The purpose is to inspire somebody to find out a more general approach.