Fuzzy implications based on Abel semigroups

At first, two innovative families of fuzzy implications are introduced. One is called an M_f-implication defined by a strictly decreasing function such that its range belongs to an Abel semigroup, the other is said to be an M-implication. After this, several proposed properties of M_f-implications and M-implications are studied. Finally, it is shown that M_f-implications and M-implications are really new class, in other words, they are indeed different from all of the known fuzzy implications.