Strong semilattices of implicative semigroups

It is well known that a strong semilattice of semigroups is still a semigroup. However, it is not known whether the analogous result carries or not for a strong semilattice of implicative semigroups. The main difficulty is that the implicative operation * is hard to define on the semilattice, especially we. need the operation * of the implicative semigroup to be compatible with the negatively ordered relation and the semigroup multiplication on the semilattice of implicative semigroups. In this paper, we provide a practical method of constructing such a strong semilattice of implicative semigroups. Our method is particularly useful in constructing implicative semigroups of large size. A constructed example is given.