Finiteness conditions for some semigroup families and constructions and efficiency

In this study some finiteness conditions for semigroup constructions are considered.Periodicity, local finiteness, residual finiteness and soluble word problem are considered for S, ? and S/? where ? is a congruence on a semigroup S . Let I be a semilattice, let Si (i ? I) be a family of disjoint semigroups, and let S=S[I; Si, Øj,i] be the strong semilattice of semigroups { Si } i ? I. Similarly, periodicity, local finiteness and local finite presentability are considered for S[I; Si, Øj,i] .Let G be a finitely presented group and N be a normal subgroup of with finite index. It is shown that the congruence?N ={(s,t): st-1 ? N } on G is finitely generated and finitely presented semigroup.