Some Finiteness Conditions for Strong Semilattice of Semigroups

Let be a strong semilattice of semigroups such that is finite and each be a family of disjoint semigroups. In this article some finiteness conditions which are periodicity, local finiteness and locally finite presentability are considered for . It is proven that a strong semilattice of semigroups is periodic, locally finite, locally finitely presented and residually finite, respectively if and only if is finite and each semigroup is periodic, locally finite, locally finitely presented and residually finite, respectively.