Equivariant Nica-Pimsner quotients associated with strong compactly aligned product systems

We parametrise the gauge-invariant ideals of the Toeplitz-Nica-Pimsner algebra of a strong compactly aligned product system over $\mathbb{Z}_+^d$ by using $2^d$-tuples of ideals of the coefficient algebra that are invariant, partially ordered, and maximal. We give an algebraic characterisation of maximality that allows the iteration of a $2^d$-tuple to the maximal one inducing the same gauge-invariant ideal. The parametrisation respects inclusions and intersections, while we characterise the join operation on the $2^d$-tuples that renders the parametrisation a lattice isomorphism. The problem of the parametrisation of the gauge-invariant ideals is equivalent to the study of relative Cuntz-Nica-Pimsner algebras, for which we provide a generalised Gauge-Invariant Uniqueness Theorem. We focus further on equivariant quotients of the Cuntz-Nica-Pimsner algebra and provide applications to regular product systems, C*-dynamical systems, strong finitely aligned higher-rank graphs, and product systems on finite frames. In particular, we provide a description of the parametrisation for (possibly non-automorphic) C*-dynamical systems and row-finite higher-rank graphs, which squares with known results when restricting to crossed products and to locally convex row-finite higher-rank graphs.