We study how decompositions of a quasi-lattice ordered group (G,P) relate to decompositions of the Nica-Toeplitz algebra and Cuntz-Nica-Pimsner algebra of a compactly aligned product system X over P. In particular, we are interested in the situation where (G,P) may be realised as the semidirect product of quasi-lattice ordered groups. Our results generalise Deaconu's work on iterated Toeplitz and Cuntz-Pimsner algebras --- we show that the Nica-Toeplitz algebra and Cuntz-Nica-Pimsner algebra of a compactly aligned product system over N^k may be realised as k-times iterated Toeplitz and Cuntz-Pimsner algebras respectively.

This research was supported by an Australian Government Research Training Program (RTP) Scholarship and by the Marsden grant 15-UOO-071 from the Royal Society of New Zealand.