An introduction to ω-extensions of ω-groups

Let ε stand for the set of nonnegative integers ( numbers ), V for the class of all subcollections of ε ( sets ), Λ for the set of isols, Λ R for the set of regressive isols, and for the set of mappings from a subset of ε into ε ( functions ). If ƒ is a function we write δƒ and ρƒ for its domain and range, respectively. We denote the inclusion relation by ⊃ and proper inclusion by ⊊. The sets α and β are recursively equivalent [written: α ≃ β ], if δƒ = α and ρƒ = β for some function ƒ with a one-to-one partial recursive extension. We denote the recursive equivalence type of a set α , { σ ∈ V ∣ σ ≃ α } by Req( α ). The reader is assumed to be familiar with the contents of [1], [2], [3], and [6]. The concept of an ω -group was introduced in [6], and that of an ω -homomorphism in [1]. However, except for a few examples, very little is known about the structure of ω -groups. If G is an ω -group and Π is an ω -homomorphism, then it follows that K = Ker Π and H = Π(G) are ω -groups. The question arises that if we know the structure of K and H , then what can we say about the structure of G ? In this paper we will begin the study of ω -extensions, which will give us a partial answer to this question.