On the extension of one-parameter operator semigroups to completions of Archimedean vector lattices

Extensions of one-parameter operator semigroups on Archimedean vector lattices to their order/ru-completions are studied. Existence and uniqueness of the extension to the ru-completion is established in the class of positive semigroups. An extension theorem for positive ru-continuos semigroups on vector lattices with property ® is proved. This theorem allows one to abandon the requirement of ru-completeness in various results concerning positive ru-continuous semigroups.