Arithmetic and other properties of certain Delphic semigroups. I

Summary Some aspects of Delphic semigroups in general — in particular, the idea of an hereditary subsemigroup, which has many uses in connexion with Delphic semigroups — are first treated. After that, attention is directed to the arithmetic of ⁺, the semigroup of positive renewal sequences. In a Delphic semigroup the aboriginal elements are the simples and the members of I₀: a class of simples of ⁺ is constructed and the simples are shown to be residual. I₀ is explicitly identified, and this leads to a canonical factorization of ⁺. The properties of division in ⁺ are discussed.