The concept of an α-space was introduced and studied by Dekker in [1] and [2] from which we take our terminology and notation. The reader is assumed to be familiar with their contents. In this paper we are interested in the following conjecture which appears in [2, p. 493]:
(*) S ⊕ C = V and S ∥ C and V an α-space ⇒ both S and C are α-spaces.
It can be simply rephrased as:
(**) The two components of an α-decomposition of an α-space are both α-spaces.
In either form it is relevant to the ideas of α-subspace and α-homomorphism. Dekker [2, T5] has established (*) in the case S (or C) is an isolic α-space. In this paper we will prove (*) in two specific cases, one of which is a generalization of Dekker's result.