This presentation gives a rough overview of some structures and results covered in my master thesis. This presentation is separated into three parts. First, the presentation briefly introduces \(C_0\)-semigroups and motivates irreducibility based on examples. The second section introduces vector lattices and generalises irreducibility. The final section uses the definitions introduced in the second section and reviews some of the introduced \(C_0\)-semigroups from the first section.