Realising the C-algera of higher-rank graph as an exel’s crossed product

We use the boundary-path space of a finitely-aligned k-graphLto construct a compactly-aligned product system X, and we show that the graph algebra C_(L) is isomorphic to the Cuntz–Nica–Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it