Idempotent closing and opening operations in fuzzy mathematical morphology

A logical approach to the fuzzification of binary mathematical morphology is presented. The fuzzy dilation and fuzzy erosion are introduced independently, using the fuzzy logical operators `conjunctor´ and `implicator´. In this way, duality relationships are not forced from the very beginning. It is shown that by choosing suitable fuzzy logical operators, all classical duality and other relationships can be preserved. Following a similar line of reasoning, it is possible to obtain the idempotence of the fuzzy closing and fuzzy opening. This important result leads to the introduction of the concept of B-open and B-closed fuzzy objects. Fundamental classical theorems are generalized for the minimum operator and its residual implicator, and for the Lukasiewicz t-norm and its residual implicator