  We study operator algebras arising from monomial ideals in the ring of
polynomials in noncommuting variables, through the apparatus of subproduct
systems and C*-correspondences. We provide a full comparison amongst the
related operator algebras. For our analysis we isolate a partially defined
dynamical system, to which we refer as the {\em quantised dynamics} of the
monomial ideal.
  In addition we revisit several previously considered constructions. These
include Matsumoto's subshift C*-algebras, as well as the tensor and the Pimsner
algebras associated with dynamical systems or graphs. We sort out the various
relations by giving concrete conditions and counterexamples that orientate the
operator algebras of our context.
  It appears that the boundary C*-algebras do not arise as the quotient with
the compact operators unconditionally. We establish a dichotomy to this effect
by examining the resulting tensor algebras. We identify their boundary
representations, we analyse their C*-envelopes, and we give criteria for
hyperrigidity. Moreover we completely classify them in terms of the data
provided by the monomial ideals. For tensor algebras of C*-correspondences and
bounded isomorphisms this is achieved up to the level of local conjugacy (in
the sense of Davidson and Roydor) for the quantised dynamics. For tensor
algebras of subproduct systems and algebraic isomorphisms this is achieved up
to the level of equality of monomial ideals modulo permutations of the
variables.
  In the process we accomplish more in different directions. Most notably we
show that tensor algebras form a complete invariant for isomorphic (resp.
similar) subproduct systems of homogeneous ideals up to isometric (resp.
bounded) isomorphisms.
  The results on local conjugacy are obtained via an alternative proof of the
breakthrough result of Davidson and Katsoulis on piecewise conjugate systems.
For our purposes we use appropriate compressions of the Fock representation. We
then apply this alternative proof locally for the partially defined quantised
dynamics. In this way we avoid the topological graphs machinery and pave the
way for further applications. These include operator algebras of dynamical
systems over commuting contractions or over row commuting contractions.
