  This paper investigates the class of finitely presented monoids defined by
homogeneous (length-preserving) relations from a computational perspective. The
properties of admitting a finite complete rewriting system, having finite
derivation type, being automatic, and being biautomatic are investigated for
this class of monoids. The first main result shows that for any consistent
combination of these properties and their negations, there is a homogeneous
monoid with exactly this combination of properties. We then introduce the new
concept of abstract Rees-commensurability (an analogue of the notion of
abstract commensurability for groups) in order to extend this result to show
that the same statement holds even if one restricts attention to the class of
$n$-ary homogeneous monoids (where every side of every relation has fixed
length $n$). We then introduce a new encoding technique that allows us to
extend the result partially to the class of $n$-ary multihomogenous monoids.
