The author introduces a notion of idempotency of orthogonal arrays (OA) of index unity, and the notion of incomplete OA. Denote by \(\text{OA}_\lambda(t,r,s)\) an array with \(r\) rows and \(\lambda s^t\) columns, with entries taken from an \(s\)-set \(E\), such that in each set of \(t\) rows every \(t\)-tuple of entries occurs precisely \(\lambda\) times. An incomplete OA is obtained if for some \(s'\)-subset \(E'\subset E\) of the set of entries each such projection onto \(t\) rows contains each \(t\)-tuple in \(E^t\backslash E^{\prime t}\) precisely \(\lambda\) times. The number of columns is then \(\lambda\cdot(s^t-s^{\prime t})\). An \(\text{OA}_1(t,r,s)\) is idempotent if it contains a chain of subarrays \(\text{OA}_1(t-1,r,s)\supset\cdots\supset\text{OA}_1(1,r,s)\). The main result (Theorem 1) of the paper is a recursive construction, which uses OA, idempotent OA and incomplete OA, all of the same strength \(t\) and with index unity as ingredients, and produces an incomplete OA with (in general) large index \(\lambda\) on a larger set of symbols. As a corollary (Theorem 2) a recursive construction for OA is obtained. This is an interesting result as it is shown that this construction contains \textit{A. Sade's} singular direct product [Ann. Soc. Sci. Bruxelles, I. Sér. 74, 91-99 (1960; Zbl 0100.02204)] as a special case. As most of the paper is concerned with OA of index unity it would have been interesting to establish a link to MDS codes. An early example, which is given explicitly in the paper, is in fact equivalent to the hyperoval in the projective plane of order 4 (and to the hexacode in coding-theoretic terminology). The link to coding theory and geometry would have been of interest. Also, some of the proofs are rather long and formal. For example, Lemmas 1 and 2 are identities involving binomial coefficients. These can be proved using basic principles of counting. The clumsy proofs of the paper are not necessary.